Probably it's more natural to talk about the Jordan decomposition and regularity when the groups are interpreted as semisimple algebraic groups (or real forms thereof). I don't think there is a special name for the homomorphisms you describe. But your question should have an affirmative answer in the algebraic group setting as an application of ideas in the paper by A. Borel and J. Tits,
*Homomorphismes "abstraits" de groupes algebriques simples*, Ann. of Math. 97 (1973), 499-571. This paper has the most comprehensive treatment of what is possible for abstract homomorphisms relative to various fields of definition, etc.

An important feature of regular semisimple elements in such an algebraic group is their *density*, whereas Borel-Tits show in effect that abstract homomorphisms are fairly close to being algebraic group morphisms that would respect semisimple and unipotent elements along with their centralizers. I'll take another look at the paper to see how close it comes to answering your question directly, but anyway it's available online through JSTOR.

ADDED:

1) Working with these groups over $\mathbb{C}$ simplifies matters a lot. For example, a connected semisimple Lie group is the same as a connected semisimple algebraic group (Chevalley), where the Jordan-Chevalley decomposition exists and is preserved under rational homomorphisms. In this algebraically closed characteristic 0 setting, the Borel-Tits study of abstract homomorphisms also simplifies and overlaps earlier papers on Chevalley groups, etc. Since Borel-Tits aim for maximum generality, their hypotheses get fairly technical and are not always needed over $\mathbb{C}$. Indeed, I'm not quite convinced that you need your hypothesis on the behavior of regular semisimple elements.

2) However, working with real Lie groups is appreciably more complicated. For example, some of these are not linear algebraic groups (making the notion of semisimple or unipotent element less obvious). Borel-Tits mostly avoid considering anisotropic algebraic groups over a field which is not algebraically closed. For semisimple Lie groups, anisotropic = compact. Fortunately however, all elements of a compact Lie group are "semisimple" (while a compact Lie group itself is algebraic over $\mathbb{R}$); so your question doesn't arise here.

3) Though it's probably not directly relevant to what you are looking at, there is a fairly long history involving real Lie groups (for instance continuity of their abstract homomorphisms), going back to work of Freudenthal and others. Following the Borel-Tits paper, Tits himself focused more directly on Lie groups in a concisely written conference article. The promised sequel with more details apparently never appeared: *Homorphismes “abstraits” de groupes de Lie.*
Symposia Mathematica, Vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome,
1972), pp. 479–499. Academic Press, London, 1974.