[1] The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory.
[2] Some of these are classification theorems of objects which are mainly dealt with in the field.
Likewise, the mathematical literature sometimes refers to the fundamental lemma of a field.
The term lemma is conventionally used to denote a proven proposition which is used as a stepping stone to a larger result, rather than as a useful statement in-and-of itself.
[3] There are also a number of "fundamental theorems" that are not directly related to mathematics: