The way that I learned what I have of the ancient grammar vocabulary was reading this Greek Appendix to Clyde’s Greek Syntax.
τὰ δὲ τοῦ λόγου στοιχεῖα λέγονται ὧδε: ἄρθρον, ὄνομα, ἐπίθετον, ἀντωνυμία, ῥῆμα, ἐπίῤῥημα, πρόθεσις, σύνδεσμος, ἐπιφώνημα.
Article, noun, adjective, pronoun, verb, adverb, preposition, conjunction, interjection.
In his preface, Clyde mentions that he uses Asopios as a guide/inspiration for adopting the ancient terminology for a modern grammatical description. For the ancients ὄνομα meant both noun and adjective, ἄρθρον both article and relative pronoun, and interjections were a type of ἐπίῤῥημα. Unlike Clyde, they also considered μετοχή (participle) as its own part of speech instead of classing it as a type of verb.
Here’s how the parts of speech terms make sense to me anyway:
ἄρθρον - lit. “a joint” was thought of by the ancients in terms its connecting force (to a previous mental conception), and you can see why it might include the relative
ὄνομα - lit. a name or label
ἐπίθετον - an epithet, and so used to mean adjective.
ἀντωνυμία - instead of an ὄνομα
ῥῆμα - the thing being said, hence a verb
ἐπίῤῥημα - something that acts on a ῥῆμα
πρόθεσις - set before
σύνδεσμος - a tying together
ἐπιφώνημα - a word thing at [a point]
μετοχή - two things happening together (ie., a verb and an adjective)