English is a funny old language isn’t it?
The confusion with the initial question may be with the word “record”.
In the title to the question, the comparison of “keep” to “record” suggests that both words are to be taken as verbs rather than nouns - as in keep (to retain) and record (long e, accent on second syllable, meaning, in this context, to write down a synopsis of a meeting - sort of.) Notice I used the word synopsis rather than that perfectly good word “record” (short e, accent on first syllable), despite record being a more widely understood word meaning nearly exactly the same thing as synopsis in this scenario.
So “record a record” of the meeting may be as grammatically correct as “take minutes” of the meeting, but it’s just bloody awkward! One would probably find that noun form of record paired off with keep if needing a synonymous phrase for taking minutes. But “keeping a record” of a meeting isn’t quite the same thing as “taking minutes” of that meeting.
You just aren’t going to find a phrase as succinct as “taking minutes”, I’m afraid. Probably because in current popular usage it means precisely one thing - that tedious and unloved task of writing down notes from a meeting for future reference. No ambiguities there. Note: while in this day and age the taking of minutes may be facilitated by a recording device in a meeting itself, there will be, at some point, some poor bloke whose task it is to transcribe a synopsis of what happened and/or was said in that meeting. That “recording” of the meeting is technically “keeping a record”, but the minute taking can’t really be said to have occurred until that transcription is done. In fact, the minute taker may not ultimately be the record keeper! They may hand the (mostly ignored) thing off to someone else whose responsibility is to retain it for future consultation. (Just ask any librarian.)
In this context “minutes” is splendidly precise! Minute suggests small, or even tiny, which leads us to hope that when we need to refer to them in the future they are brief - a short summing up in point form. Synopsis is about the only other word that I can think of that comes close, but it is a bit more vague - minutes and meetings go together like PB&J.
And “take”? Well there isn’t really a suggestion of writing stuff down until “minutes” is paired with it. Just like “keep records” doesn’t have a writing down feel to it like “keeping a diary” does. Too many alternate forms of records these days, while diaries are still pretty much written manually with hearts over the Is by thirteen year old girls. Or typed into a laptop by scientists and would-be celebrities planning on future memoirs.
So there you go. “Keep” and “record” don’t really mean the same thing. And together they just can’t beat “taking minutes”.