Mind you, some authors – e.g. D.J.H. Garling in A Course in Galois Theory (1986, Cambridge University Press) – insist that a ring homomorphism should map the multiplicative identity of one ring to that of the other. If you adopt such a definition, the second possibility would not be possible: all homomorphisms from a field to a ring would then have to be injective.
I like to distinguish between "homomorphism between rings" and "homomorphism between commutative unitary rings". So when I see "ring homomorphism" all I think of is . However, if I see "commutative ring homomorphism" (it is rather common to called commutative unitary rings simply by commutative rings) then I think of the additional condition .