Dear all,
It is worth keeping in mind that (an)at/(an)adanta etc. are entirely formal in Pāṇini's grammar and have been rightly translated as "that which ends (or does not end) in shot a." What specifically ends (or does not end) in short a must be ascertained from the larger context of the sūtra. For example, P. 7.1.5 ātmanepadeṣv anataḥ prescribes that at (and not ant) be substituted for jh in jha, the suffix for the 3rd person plural ātm. (or 1st person if we wish to be Pāṇinian) given in P. 3.4.78. anataḥ, "of that which does not end in short a," modifies aṅga, "stem," (not specified to either a verbal or a nominal form), which is continued from P. 6.4.1 aṅgasya. This is how Pāṇini formulates a rule that one might give to students as: in the non-thematic verbal classes, there is no n in the 3rd person pl. ātm., as apposed to labhante, padyante, etc. of the thematic classes. But Pāṇini's rule extends beyond the present tense stems and in fact should not be strictly associated with anything other than "that which does not end in short a" because it is by this same rule that aorist forms are distinguished as well: adikṣanta vs. aneṣata, etc. If we wish to use thematic and non-thematic as a sort of equivalent for Pāṇinian (an)at/(an)adanta, then it is crucial to include all places in the verbal paradigm where being (an)adanta comes into play. On the other hand, just a few sūtras later, at is used in reference to a nominal form in P. 7.1.9 ato bhisa ais, in which case at (= adanta) again modifies aṅga but one that is now nominal. The rule prescribes the substitution of ais for bhis, the latter of which is the basic ending for the instrumental plural (tṛtīyā vibhakti), hence devaiḥ and not devabhiḥ*. In any case, at/adanta is neither restricted to nominal or verbal forms nor the present stem.
I find Abhyankar's entry on adanta somewhat misleading, as least as quoted, and not terribly informative of how (an)at/(an)adanta is used in in the Aṣṭādhyāyī. The business about "roots of the tenth conjugation which are given with the letter a at their end which is not looked upon as mute (it)..." is a bit recherché and refers to a list of roots in the curādi gaṇa of the Pāṇinian dhātupāṭha. In Liebich's edition it begins on p. 190 with the head adantāḥ ita ā gaṇāntād ... The reason, I believe, that he gives slightly obscure references is that he was looking for specific instances of adanta in the Aṣṭādhyāyī and not at, which, however, is usually to be understood as adanta. The opening sentence of his entry, nonetheless, repeats more or less the translation that has already been said: "ending with the short vowel a".
All systems of grammar are fictional, and in teaching Sanskrit or whatever language, the terminology and categories that make the most sense in a specific context should be used. Comparison of preexisting systems, such as that of Pāṇini (we should also recall that there are other systems with different terminology and different definitions for the same Pāṇinian terms), should only be undertaken once the object of comparison is well understood and beginning Sanskrit is usually not the place to begin such a comparison. I do, however, support familiarization with Pāṇinian terminology but as it is defined by Pāṇini himself, not in reference to Latinate terms.
All the Best,
Victor