I am really bummed out to find that the term "strict monomorphism" is already used to mean something else.
Can anybody console me with the knowledge that there is another name I can use for a monomorphism that is not an isomorphism?
I have observed that "proper monomorphism" is sometimes used to mean "a monomorphism that is not an isomorphism".
For example, if $A$ is a subset of $B$ and if $A\neq B$, then we write that "$A$ is a proper subset of $B$". We know in the category of sets, for example, that if there is an monomorphism from $A$ into $B$, then $A$ can be identified with a subset $C$ of $B$. In this case, $C$ is a proper subset of $B$ if and only if $f$ is a "monomorphism that is not an isomorphism".
However, I could be incorrect as I do not have a definite source for this terminology.
You could go with "non-invertible monomorphism", which is really just a restatement of "monomorphism that isn't an isomorphism". But as far as I know, there is no "special" or "reserved" name for such maps.