Dear Prof. Jeffrey Long,
'existence' , interestingly enough, is part of the contemporary neologisms in the discourse related to religion in the contemporary Indian languages.
Words aastka and naastika are the neologisms formed and in use as the Indian language replacements for 'theist' and 'atheist' of English.
But , as almost all in this group know, these two words
aastka and naastika were not in reference to the existence or no existence of God in the traditional Vaidika discourse.
asti and naasti, in the traditional Vaidika discourse , refer to the existence and no existence of praamaanya for Sruti or existence or no existence of paralokas , svarga, naraka or mokshalokas (such as vaikuntha, goloka, kailaasa, manidveepa etc.)
The root sat for exist as found in the word satya is part of the debates within Vedanta about the satyatva or mithyaatva of the category called Jagat , but not about a category comparable to God.
Category comparable to God, or a category often conflated with or confused for God in the Vaidika darshanas including Vedanta, is " eeshvara ".
Classificational categories seshvara and nireeshvara used in reference to different Vaidika darshanas are not about the existence or no existence of
" eeshvara ". It is about whether the category of
" eeshvara ". is necessary to account for things , (entities, processes and relationships etc.)
Within Vedanta, brahman and eeshvara , categories comparable to God, are not discussed for existence or no existence of the category/entity.
Another existence or no existence discussion within Bharatiya Darshanas seems to be around aatman and that is between Vedanta and an avaidika darshana, say Baudha darshana . If anattaa is no existence of aatman, then this can be seen to be around the question of existence.
In Vedanta, because of the equivalence of Brahman and aatman at a certain level, this, if it is actually around the existence or no existence of aatman, can be seen to be, at least remotely, connected to the existence or no existence of a category comparable to God.