Ok, might as well put in my 2 cents...

My first conlang was originally just referred to by the name of the
universe, which was the word "to be", or Ea (stolen from The Silmarillion,
where I think it was the universe-creating word).  My files on the
language tend to start "lanea" for language of ea, so I also think of it
as lanea sometimes.

But it does have an official name.  One evening in high school I made
up a name for the nebulous, alleged people who speak the language (I'm
not good at the conculture side of things - I had no concept of the
human side of the equation, as evidenced by my abstruse and nearly
useless panthea-on).  The people were called the mėrčchet, and their
language was then mėrčchi.

The second language I named Toma Efim, which means "My Language."  It
is today known as Toma Heylm, which means "Our Language", on the theory
that it, too, must have a hypothetical body of speakers.

One sketch in which I was exploring ways of creating subclauses in an
ergative context will forever be known to me as "Chicken-fox-farmer
lang", because the example sentences concerned a chicken, fox, and farmer.
It has grammar only, no vocabulary; the example sentences were English

The verbless language sketch, tekem, is known by its own word for
"language".  The not-yet-quite-created alien tonal language has a working
title of Disharmony, which I hope will not be the result.  I can't
remember if the impenetrable sandhi sketch from about 1996 had a name.
I don't think my high-school attempt at an all-verb language did, though
I did create a word for "language" in it while trying to come up with an
entry for the first conlang t-shirt project (the "your language here" one).

But basically, my languages are all named "language".  I don't believe
this presents a problem in terms of ambiguity because the speakers of
my language are unlikely to be discussing it, and if they do, they really
won't need to get any more specific than "our language" vs. "their
language".  But I did like the one person who mentioned that his con-
universe was named after "purple", for which he'd back-formed a philosophical
justification.  That was creative.