I think one of the reasons (not the only reason) why we only talk about "exegesis" in biblical studies and don't use "exegesis" in other fields may be related to the reason why we only say "baptism" to refer to the ritual but don't say "I baptized my bread in the oil" when we're at Olive Garden. Both are loanwords from Greek that we came to use in certain situations because of their specific association with a specific semantic domain in Greek from which the loan came. In other words, we took these loanwords from Greek not in their general sense, but only in a specific realm. So, we took βαπτισμα for the ritual and εξηγησις for the explication of scriptures.
In the ancient sense, εξηγησις seems to simply mean explaining, and in the context of the church fathers, often explaining the scriptures:
Epiphanius, after explaining a passage, concludes with (Panarion, 2.476.5):
ἀλλὰ γὰρ περὶ τῆς κατὰ τὸν ψαλμὸν ἐξηγήσεως ἅλις ἔχει ταῦτα.
'For these things (said above) are sufficient for the ἐξήγησις of the psalm.'
Chrysostom about Paul (de Sacerdotio, 4.7.20–21:
οὐχ ὅλας ἡμέρας καὶ νύκτας ἀνήλισκεν ἐφεξῆς εἰς τὴν ἐξήγησιν τῶν Γραφῶν;
'Was he (Paul) not absorbed whole days and nights continuously in the ἐξήγησις of the Scriptures?
I think the issue is with those who take the term "exegesis" to mean a hyper-grammatical analysis of every term and translation, though I do not think it has to mean that in the modern sense. I think someone who uses the term "exegesis" or "exegete" may simply refer to explaining and teaching the scriptures. It seems to me that the reason we talk about "exegesis", may be that the tradition of interpreting/explaining the scriptures grew mostly in Latin and Greek, and that was one term they made use of for it. The fact that it has come to be associated with hyper-grammatical readings does not mean that the use of "exegesis" in biblical studies—when we don't use it for other fields—means that it has to be associated with hyper-grammatical readings. I think its uniqueness for biblical studies may have more to do with it being a loan from Greek/Latin in that specific field. There are all sorts of words that we only use in biblical studies for things like that that we don't use in other fields: expositional preaching (simply means teaching through a book instead of topical, classicists also write commentaries for entire texts but I've never heard a classicist use the word "expositional" to refer to this).
Long story short, while I agree with everything in principle that Randall Buth and Daniel Streett say about exegesis, I think it might be worth considering the nuances of the history of how this word came to mean what some think it to mean, and that someone may simply mean "explaining" when they use the word "exegesis" as well (I hope).
Also, while I agree with a previous commenter that the church fathers never engaged in that hyper-grammatical analysis, they certainly would explain the scriptures with grammatical comments ...
Nilus Ancyranus (Εἰς τὸ τῶν Ἀισμάτων Ἄισμα, 80.6–7)
μητέρα δὲ τὸ πνεῦμα λέγει τὸ γεννῶν τοὺς βαπτιζομένους καὶ θαυμάσει μηδεὶς εἰ θηλυκῷ ὀνόματι καλεῖ τὸ πνεῦμα. (7) ἡ γὰρ σχέσις τῶν γεννωμένων ἀπαιτεῖ τοῦτο καὶ ἡ Ἑβραϊκὴ δὲ φωνὴ θηλυκῶς ὀνομάζει τὸ πνεῦμα διὰ τὴν αὐτὴν αἰτίαν ...
'And he says that the Spirit is a mother, who begets those who are baptized and no one will marvel if he calls the spirit with a feminine name/noun, for the relation of those who are begotten requires this and the Hebrew idiom calls the spirit by the feminine gender for the same reason ...