What do you call an embedding (or map in general) from a manifold to a manifold $i: M \rightarrow N$, such that $i(\partial M) \subseteq \partial N$?
Is there a difference, if we define it as a relative map $i: (M,\partial M) \rightarrow (N,\partial N)?$
There is a phrase from Hirsch's book "Differential Topology" which applies to a very closely related situation.
Namely, if the inclusion $i(\partial M) \subset \partial N$ is strengthened to an equation $i(\partial M) = i(M) \cap \partial M$, and if for each $x \in \partial M$ the subspaces $D_xi(T_x(\partial M))$ and $T_{i(x)} (\partial N)$ are required to be transverse subspaces of the vector space $T_{i(x)} N$, then one says that $\text{image}(i)$ is a neat submanifold of $N$. I suppose one can also say in this situation that $i$ is a neat embedding of $M$ in $N$.
This is definitely a stronger property than what you stated, but one might argue that it more accurately reflects the intuition of "respecting boundaries".