Background
I had originally posted this question as:
Among morphisms there are homomorphisms that are structure-preserving maps between algebraic structures of the same type. What's a conventional term for a structure-preserving map between algebraic structures of different types?
However it has been pointed out to me that this original question is flawed in its assumptioned. To salvage this question into something coherent, and valuable to users of math.SE I will give some explanation of why I found this terminology confusing.
Why is this terminology confusing
I heuristically use patterns in terminology to either predict, better understand, or remember terminology. This can lead to false inferences sometimes. An example is when there's a prefix like homo-, there is often an antonymous prefix hetero-. When something is worth using the prefix for same, it has often suggested that there is a corresponding prefix for other or different.
In statistics we contrast homoscedasticity from heteroschedasticity. In biology we distinguish heterostasis from homostasis. In chemistry we distinguish homogenous mixtures and heterogenous mixtures.
Not to say that we should upend the current state of mathematical terminology, but I hope this post helps someone else with this gotcha of mathematical jargon.
No, a homomorphism is not a mapping between two algebraic structures of the same type - it's not clear what "type" would mean anyway. A homomorphism is a structure preserving map between two algebraic structures. Specifically, if we define "algebraic structure" to mean "set with a binary operation on it", then a homomorphism is a mapping from $(A, \cdot)$ to $(B, \circ)$ satisfying
$$f(x\cdot y)=f(x)\circ f(y)$$
A homomorphism is not a more specific type of "morphism", homomorphism and morphism are just two different words used in two different fields of math (homomorphism in algebra and morphism in category theory).
Now, if we have in mind a specific kind of algebraic structure like a group, we might talk about a "homomorphism of groups" to refer specifically to refer to a homomorphism between groups, but that doesn't mean you're not allowed to use the term homomorphism to refer to a mapping between any two algebraic structures.