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I’ve usually heard “finitely generated” used to refer to the property you call “finite type”.
A little googling suggests that “finite type” may be more commonly used for ring homomorphisms, to distinguish finite generation as an algebra from finite generation as a module. If they mean essentially the same thing for modules, then it doesn’t seem like a good idea to me to arbitrarily decide to generalize them differently to the abstract context.
This might be a good question for mathoverflow, or the categories mailing list?
A ring map of finite type means the codomain is finitely generated as an algebra over the domain, though… http://stacks.math.columbia.edu/tag/00F3
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