I suspect there are, but I think I should perhaps ask: are there extant 
languages which divide non-Personal tenses into "animal" and "other" ?

an example of what I'm looking for a name for the phenomenon, is at  for example:

word // Moghul translation // tense // example

*  have  // qa"an  //  Non-Person Possessing, Past & Present Tense  //   The 
boulder has lichen (on it), The boulder had moss growing (on it), The book 
is damaged, The book has damage (to it).

* have  // qa"ana  // Non-Person Possessing, Future Tense    //  The boulder 
will have moss on it.

* have, attached-to  // qeda=  // Animal Possessing, Present Tense  //   The 
horse has a leech (on it).

* have, attached-to  // qe"eda  // Animal Possessing, Past Tense    //  The 
horse had a leech (on it)

Thank you all for your time and patience.  Any & all replies are welcomed.