Two definitions of strong homomorphism

I can't answer your first question, but I can assure you that the definition you know (a strong homomorphism preserves and reflects relations) is more standard!

A small correction: You say "the second definition is just the first one plus requirement that the image of $f$ contains all related elements of $B$". It does require this, but it's also much weaker in its reflection requirement, only requiring the some preimage of related elements in $B$ be related in $A$, not all.

I've never seen the definition from Wikipedia before, and I agree that it seems very unnatural. I suspect it might be a typo, but I'm not a expert in general relational structures.

I can tell you one thing (either definition of) strong homomorphisms are good for - showing that surjections are orthogonal (see http://ncatlab.org/nlab/show/orthogonality) to injections.

Just to be clear, let's call a homomorphism strongly reflective if

$R^B(f(\overline{a}))\Rightarrow R^A(\overline{a})$,

and weakly reflective if

$R^B(f(\overline{a}))\Rightarrow \exists a':R^A(\overline{a'})\wedge f(\overline{a'})=f(\overline{a})$

(so this is just like the Wikipedia definition, without requiring surjectivity, which I agree seems unnatural). Then you can show that if $e$ and $m$ are surjective and injective weakly reflective homomorphisms respectively, they are orthogonal in the following sense - whenever $ve=mu$, for any other weakly reflective homomorphisms, there exists a (unique) weakly reflective homomorphism $d$ with $de=u$ and $md=v$. One important consequence of this is that a weakly reflective homomorphism that is both injective and surjective must in fact be an isomorphism. This can fail for arbitrary homomorphisms - the identity map from $\mathbb{R}$ with the discrete order (i.e. $x\leq y\Leftrightarrow x=y$) to $\mathbb{R}$ with its usual linear order is an injective surjective homomorphism (of structures with a single relation $\leq$) but certainly not an (order) isomorphism.

Of course, the same orthogonality result holds also for strongly reflective homomorphisms, but you may not want to throw away any homomorphisms that you don't have to. For example, the (quite useful!) coordinate projections on a product $\mathbb{P}\times\mathbb{Q}$ of partially ordered sets $\mathbb{P}$ and $\mathbb{Q}$ are weakly reflective but not strongly reflective.