Why injection into sum type apparently leads to ambiguity?
I think the key point here is $\sigma$, $\tau$ and $\phi$ are type variables, and not specific types. So, what the typing rule for $\operatorname{inl}$ says is that $\operatorname{inl} M$ is of type $\sigma + \tau$ for any $\tau$. The names of type variables don't matter, what matters is where else are you using the same type variable.