My apologies for not having been clear enough in my original post.
I’d like to hit the Reset Button Hilary Clinton has been carrying around.

All of this “n-ptote” business has to do with the number of distinct case-
inflections (e.g. case-endings) on the noun or substantive or pronoun or 
adjective in question.  Different locations in the clause, and different 
adpositions, don’t count.
For convenience I’m going to call them “case-endings”, though technically 
they don’t have to be endings; they could be prefixes, or ablaut, or tonemes, I 
suppose.

It’s only about those inflections and that morphology which communicates 
case, possibly fused with something else such as definiteness or number.

It’s only useful in languages in which some forms have a certain number of 
case-endings and other forms have a different number.


What dictionaries have to say about these words:

According to the Compact Oxford English Dictionary Second Edition ISBN 0-19-
861258-3:
Aptote was first used in 1589
Monoptote in 1612
Diptote in 1612
Triptote in 1612
Tetraptote in 1656
Pentaptote in 1656
Hexaptote in 1658
Polyptote in 1656.

In 1885 in “Socin’s Arab. Gram.”, it was used thus: “The triptote  are 
distinguished from the diptote  nouns by the nunation always being written 
over the former.”

OED defines
Aptote an indeclinable noun
Monoptote a noun occurring in a single oblique case
Diptote a noun that has only two case-endings
Triptote a noun or other word having only three case-endings
Tetraptote a noun with only four cases
Pentaptote a noun with five cases
Hexaptote a noun declined with five cases
Polyptote a noun that is declined with many case-endings

(Actually, there’s some ambiguity about polyptote, depending where the 
morpheme-boundary goes n relation to the second <p>.  It could be the sum 
of the census of the nematocyst-bearing coelenterates, say; or a large open-
mouthed two-handled bag for carrying nasal tumors.)

(OK, that part wasn’t serious.  What is serious is, “polyptote” is sometimes 
used as a synomym for “polyptoton”, a stylistic scheme of which “heart’s 
heart” and “ownest own” are examples.)

Webster’s dictionary from 1913 says:
Ap´tote    (&#259;p´t&#333;t)
n.	1.	(Gram.) A noun which has no distinction of cases; an 
indeclinable noun.
Webster's Revised Unabridged Dictionary, published 1913 by C. & G. Merriam 
Co.

Mon´op`tote
n.	1.	(Gram.) A noun having only one case.
	2.	A noun having only one ending for the oblique cases.
Webster's Revised Unabridged Dictionary, published 1913 by C. & G. Merriam 
Co.

Dip´tote
n.	1.	(Gram.) A noun which has only two cases.
Webster's Revised Unabridged Dictionary, published 1913 by C. & G. Merriam 
Co.

Trip´tote
n.	1.	(Gram.) A noun having three cases only.
Webster's Revised Unabridged Dictionary, published 1913 by C. & G. Merriam 
Co.

Tet´rap`tote
n.	1.	(Gram.) A noun that has four cases only.
Webster's Revised Unabridged Dictionary, published 1913 by C. & G. Merriam 
Co.

Pen´tap`tote
n.	1.	(Gram.) A noun having five cases.
Webster's Revised Unabridged Dictionary, published 1913 by C. & G. Merriam 
Co.

Note that neither the OED nor Websters Revised Unabridged 1913 have the 
word “only” in the definition of <pentaptote>, though they both include it in 
the definitions of mono- and di- and tri- and tetra-ptote.

Here are what some other online dictionary-like sources say about these 
words’ definitions:
Wiktionary, the free dictionary
diptote (plural diptotes)
1.	(grammar) In grammar, a noun which has only two cases.
triptote (plural triptotes)
1.	(grammar) In grammar, a noun which has only three cases. 
http://dictionary.reference.com/browse/
Ap"tote\ ([a^]p"t[=o]t), n. [L. aptotum, Gr. ? indeclinable; 'a priv. + ? fallen, 
declined, ? to fall.] (Gram.) A noun which has no distinction of cases; an 
indeclinable noun. 
Webster's Revised Unabridged Dictionary, © 1996, 1998 MICRA, Inc. 
Mon"op*tote\, n. [L. monoptotum, Gr. ?; mo`nos single + ? apt to fall, fallen, 
fr. ? to fall; cf. ? case.] (Gram.) 1. A noun having only one case. --Andrews. 
2. A noun having only one ending for the oblique cases. 
Webster's Revised Unabridged Dictionary, © 1996, 1998 MICRA, Inc.

dip&#8901;tote
&#8194;   &#8194;&#712;d&#618;p to&#650;tShow Spelled Pronunciation [dip-toht]  Show IPA 
–noun 
a substantive declined in only two cases, esp. when occurring in a language in 
which this is less than the normal number.
________________________________________
Origin: 
1605–15; < LL dipt&#333;tos noun with only two cases < Gk, equiv. to di- DI- 1 + -
pt&#333;tos, verbid of píptein to fall; cf. pt&#333;sis accidence, lit., falling 
Dictionary.com Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2009. 

Trip"tote\, n. [L. triptotum, Gr. ? with three cases; ? (see Tri-) + ? falling, fr. ? 
to fall.] (Gram.) A noun having three cases only. 
Webster's Revised Unabridged Dictionary, © 1996, 1998 MICRA, Inc. 

Tet"rap*tote\, n. [L. tetraptotum, Gr. ?.] (Gram.) A noun that has four cases 
only. --Andrews. 
Webster's Revised Unabridged Dictionary, © 1996, 1998 MICRA, Inc.

Pen"tap*tote\, n. [L. (pl.) pentaptota. Gr. ? with five cases; ? (see Penta-) 
+ ? falling.] (Gram.) A noun having five cases. 
Webster's Revised Unabridged Dictionary, © 1996, 1998 MICRA, Inc.

Among things to notice is, there are two definitions for monoptote in the 1996 
Webster’s though only one in the 1913.
And among other things to notice is, in the definition of “diptote”, the 
word “substantive” is used.


Other terms:

“oblique case” and “direct case”.
One definition the OED gives for “direct” is “not oblique”.
For “Oblique” definition A.5.a is as follows:
Oblique A. adj. 5. gram. A. Any case except nominative and vocative; or, any 
case except nominative, vocative, and accusative.

I think it has been well-established for some time that one permissible usage of 
the term “direct case” includes the case for the direct object or primary 
object, in languages which have case and which have such a grammatical 
relation as “direct object” or “primary object”.

If you object to using the term in any way other than translationisms from 
Latin grammarians’ grammars of Latin, well, rest assured, you are far from 
alone in that.  But enough grammarians also include accusative (at least when 
talking about certain languages) that the makers of the OED decided to do like 
Canute and Mohammed, and not try to command the tide not to come in nor 
the mountain to move.

Also, consider the parallel with the term “substantive”.
In Latin grammarians’ Latin grammars of Latin,  what we now call “nouns” were 
called “nomina substantiva”, and what we now call “adjectives” were 
called “nomina adiectiva”.
Nowadays the use is a bit different.
In many languages many adjectives can be used as if they were nouns, and 
many nouns can be used as if they were adjectives.  Exempli gratia, “poor” 
and “rich” and “ugly” are “adjectives used as nouns” in:
“The poor you have always with you”
“In any society the minority which is least-understood is the rich.”

“Beauty’s only skin-deep;
Ugly’s to the bone.
Beauty dies, and fades away,
But ugly holds its own.”

And “sea” and “army” are “nouns used as adjectives” in 
“I was talking about the sea captain, not the army captain.”

In some languages all or almost all common nouns can be used almost exactly 
like adjectives in all or almost all places an adjective could be used; and 
likewise all or almost all adjectives can be used almost exactly like common 
nouns in all or almost all places a common noun can be used.  This is true to 
such a degree, in some of these languages, that it is useless and misleading to 
claim “common noun” and “adjective” are really two different word-classes.  
Therefore only one word-class is used for such words; and they are 
called “substantives”.  (Proper nouns are still called “nouns”.)

But a purist could complain that “noun” ought to be the general term 
and “substantive” ought to be the narrower term opposed to “adjective”.  
Nearly nobody professional does so, though, and nobody professional pays 
much attention to those who do so.

A person could point out that the pre-grammatical meaning of “substantive” 
was something like “able to stand alone” or “able to stand unassisted”.  So it’s 
not unreasonable, one might claim, to point out that an “adjective” which can 
be used alone – like “the poor” or “the rich” or “the good” or “the true” or “the 
beautiful” – is a “substantive adjective”.  
But “substantive adjective” is an oxymoron.

So I think it’s the better part of valor to just grit your teeth and accept that 
some people are going to mean “vocative, nominative, or accusative” 
whenever they say “direct case”.


Some people don’t think grammatical relations – not even subjects -- actually 
show up at all in any language.  Some don’t think they’re really very important 
or useful in any language.  Some think they do show up in some languages but 
not others.  Some think they’re important in some languages but not others.  
Some think every language has at least one grammatical relation – that every 
language has, at the very least, subjects.

If you accept that a language with cases has some grammatical relations, 
then you probably accept the notion of a “core argument” or “core term” 
or “direct core argument” or “direct core term”.  A “core argument” is a 
participant that occupies a grammatical relation.

Some languages appear to have only one GR – the Subject.
If a language has more than one GR, all except the Subject are called Objects 
(according to Trask).
Some languages have only two GRs (Subject and just one Object).
If a language has more than two GRs (more than one Object), one of the 
objects is called the Primary Object and the others are called Secondary 
Objects.  In some languages, the primary object is called “the direct object” 
and any secondary object is called “an indirect object”.
If a language has such a GR as an Indirect Object, and has cases, then I had 
intended to call the case for Indirect Objects a “direct case”.  Maybe instead I 
should have called it a “core case”; then, the “core cases” are; Nominative, 
Accusative, and Dative in languages with three GRs, but just Nominative and 
Accusative in languages with only two GRs.
One way that core cases are distinguished from “oblique cases” is that an 
addressee cannot deduce the “deep case role”, “semantic role”, “thematic 
role”, or “theta role” of the thus-cased noun-phrase (or other 
nominal/pronominal/substantival) from its case; one has to also consult the 
grammatical Voice of the verb to do so.


I have borrowed and read and returned Barry J. Blake’s book “Case”.  I am 
trying to use his terminology here; I may have messed it up.
Blake called the core cases and the genitive case “grammatical cases” or 
a “syntactic cases”.  These cases, in his opinion, mostly show how the thus-
cased word is grammatically-related or syntactically-related to some other 
constituent of the clause, rather than mostly showing how the thus-cased 
word is semantically related to some other constituent of the cause.
So, in a language with cases, with three GRs, and with nominative-accusative-
dative alignment, the “grammatical cases” would be the Nominative, the 
Accusative, the Dative, and the Genitive.

Most “oblique” cases, or at least most oblique adverbial cases, are “semantic 
cases”, in the sense that those case-“endings” are mostly used to 
communicate how the thus-cased word is semantically related to some other 
constituent; it shows the addressee what the “deep case role” or semantic 
role or thematic role or “theta-role” of the thus-cased word is.  And 
furthermore, most cases that communicate the thus-cased nouns semantic 
relationships, are “oblique”.
I did not intend at all to mean the “underlying deep case-role” when I 
said “semantic case”.  I meant instead, a case whose chief use and purpose is 
to show the theta-role of the word to which it is applied.


So;
I think it’s worth asking whether, when a noun (or other word) has just 
one “ending” for two cases (that is, these two cases are homophonous for 
that noun), is it likelier that the two cases are both “core cases” 
or “grammatical cases”?  In the familiar types of languages, these would be, 
for instance,
Nominative = Accusative or
Accusative = Dative or
Dative = Genitive or
Ergative = Genitive.
If that’s not likeliest, where does it rank in likelihood in comparison to other 
possibilities?

I think it’s worth asking whether, when a noun (or other word) has just 
one “ending” for two cases (assuming that many “nouns” in the same language 
have different “endings” for those two cases), how often is one a “core case” 
and the other an “oblique case”?
For instance,
Nominative = Oblique
Accusative = Oblique
Dative = Allative (if the language has an “indirect object” GR, so “dative” is a 
core case) 
Dative = Ablative
Dative = Benefactive
Genitive = Partitive (an example of “grammatical case with semantic case”, but 
not of “core case with oblique case”.)
If I can’t find out “how often”, I’d like to know whether it is more or less 
common than some other possibility.

I think it’s worth asking whether, when a noun (or other word) has just 
one “ending” for two cases (assuming that many “nouns” in the same language 
have different “endings” for those two cases), how often are both of 
them “oblique cases”?
For instance
Comitative = Instrumental
Instrumental = Perlative