Great question. There's a subtle point the article glosses over "... with coefficients in K." If you have two elliptic curves that just happen to be cyclic groups over finite fields, you can construct homomorphisms between them with finite kernels. But, you won't have any hope of finding morphisms unless those finite fields are the same. (same characteristic? It's been a while)

In less fancy language: isogenies are homomorphisms you can describe with simple formulas involving (x, y) coordinates.

Good question, no there is no difference for elliptic curves, which you can think of as 1-dimensional geometric objects (curves) which posses group structure. A good map in this category should respect the geometry (be a so called rational map, ie defined by polynomials in a suitable coordinate system) and the group structure. Interestingly all these maps are either constant (map everything to 0) or surjective.

For higher dimensional geometric groups (abelian surfaces etc) one usually wants to make a distinction and calls the surjective homomorphisms with finite preimage isogenies.