I thought I understood this term, but when I tried to verify this I found three different and conflicting definitions, none corresponding to mine. Is there a generally agreed definition for this term (and since it can't have all four meanings, what would be terms for the other cases) ?
Ref(1): Basic Set Theory By Nikolai Konstantinovich Vereshchagin, Alexander Shen
A mapping f: A → B has finite support if it equals the least element in B for all but a finite subset of elements of A. Requires obviously that B have some form of order and a least element.
Ref(2): Wiki: Suppose that f : X → R is a real-valued function whose domain is an arbitrary set X. The set-theoretic support of f, written supp(f), is the set of points in X where f is non-zero
Ref(3): What Does it Mean for a Function to have Finite Support?
It should mean : the function vanishes outside a set of finite measure not that only finitely many elements in the domain produce a nonzero value for the function.
(4): My own understanding
f: A → B has finite support if its domain is a finite subset of A.
The term "finite support" often appears in probability theory. Let me try to explain this concept in your preferred language.
Let $f:A\to B$ be a specific function and there exists a zero in $B$,
then, $f$ has finite support if $f(a)=0$ for all $a\in A\setminus X$, where $X$ is a finite set.
So your definition $(4)$ is not accurate enough, mathematically. In fact, the function's domain is still $A$.
The first three definitions mean similar things and are all correct.