Is there any standard name for this concept that is weaker than local one-to-one-ness?
In some open neighborhood of $x_0$ there is no point $x\ne x_0$ such that $f(x)=f(x_0)$.
Or, if you like: In some neighborhood of $x_0$, for every point $x$ in that neighborhood, $f(x)=f(x_0)$ only if $x=x_0$.
Might one simply say that "$f$ is weakly locally one-to-one at $x_0$"?
Trivial question, it might seem, if it's only about mathematics. Maybe it's about psychology of learning mathematics. I recently came across this error: If $f\;'(a)>0$, the $f$ is strictly increasing in some neighborhood of $a$. (Much less recently, I made that mistake myself, when, as an undergraduate, I was frustrated by my inability to write what I thought must be a simple $\varepsilon$-$\delta$ proof of that proposition.) Would people be less likely to make that mistake if somewhere in the recesses of their brain they subconsciously remembered hearing that "If $f\;'\ne 0$ at a point, then $f$ is weakly locally one-to-one at that point."?