Yeah, ordnance people generally work on the guns and ammunition rather than shoot them, but that was my best guess. The phaser crews are kind of an anomaly anyway, because other times it seems like pushing a button on the helm console does everything.
I don't think the term "gunnery" was ever used in TOS, but "weapons department" was. Of course in the '50s and '60s the US Navy was moving from ships having gunnery departments with guns to weapons departments with guns and missiles, but I don't know if that would have made its way in to Star Trek.
Interesting! I am not nearly as familiar with S1 as I am with S2 and S3 (on a relative scale, meaning I haven't watched a lot of first season episodes 50-100 times and don't have huge snippets of dialogue I can remember from S1 eps with some exceptions), but I thought there was one episode, probably in S1, where someone, maybe Kirk, makes a comment or issues an order about "phaser gun crews."