Yes — a meaning that fits. Since a finite field is, by definition, finite.
Why would having an entirely neutral name be better? A finite field isn’t a field that is in some sense finite or related to or inspired by finiteness in some mysterious way, it is quite literally finite (in the sense that all mathematicians use the word).
The danger of using existing terms like ‘category’ for new concepts is that they encourage the importation of lots of likely irrelevant associations and intuition that is likely to confuse rather than aid one’s understanding.
I definitely think your argument goes too far if you think the phrase ‘finite field’ is potentially misleading. You might as well demand that the phrase ‘multiples of 5’ be replaced with ‘fluffies’ because it has less meaning.
