Basically, if one is trying axiomatize ZFC, then all these versions of replacement are equivalent modulo the other ZFC axioms, and there is really nothing going on here. The axiom is quite robust and is invariant under these kind of changes. There aren't really any hidden technicalities that cannot be easily overcome. Replacement is equivalent to collection, and for collection, it suffices to say merely, of every $\phi$, that for every set $A$, there is a set $B$ such that for every $a\in A$, if there is a $b$ with $\phi(a,b)$, then there is a $b\in B$ with $\phi(a,b)$. Thus, $\phi$ need neither be total nor functional.
Meanwhile, however, if one is working in a weaker theory, such as ZFC-, meaning ZFC without the power set, then the strength of the axiom becomes sensitive to how it is formulated. For example, without the power set axiom, the replacement axiom scheme and collection are no longer equivalent. This is due to Andrej Zarach, and you can find a proof in my paper What is the theory ZFC without power set? with Gitman and Johnstone. Without the power set axiom, the situation is that replacement is much weaker than you expect. For example, in the version of ZFC- formulated with replacement, one cannot prove that $\omega_1$ is regular, or the Los theorem on ultrapowers and many other things that are provable if one uses collection instead of replacement. Our summary conclusion is that ZFC- is properly formulated with collection instead of replacement, in order for it to prove the things usually desired in this theory.
