We are talking about a theory in the language of ordered rings, so a homomorphism should respect the elements of the language:
- $ \varphi(0) = 0 $
- $ \varphi(1) = 1 $
- $ \varphi(-x) = -\varphi(x) $
- $ \varphi(x+y) = \varphi(x) + \varphi(y) $
- $ \varphi(xy) = \varphi(x) \varphi(y) $
- $x < y \implies \varphi(x) < \varphi(y) $
(I assume the language has the $<$ predicate rather than the $\leq$ predicate)
In some contexts, one might require more from a homomorphism: that it preserve something related to completeness.
An isomorphism, of course, is a homomorphism that has an inverse homomorphism.
Incidentally, note that there is only one ring homomorphism between any two complete ordered fields, and everything else follows from the ring structure (e.g. the nonnegative elements are precisely the squares), so the answer to your multiple choice question turns out to be "all of the above".