What is induced matching, maximum induced matching and mim-width?
Both definitions are the same. An induced matching $M$ in $G$ consists of a set of edges such that $G|_{V(M)} = M$, that is, the subgraph of $G$ induced by the vertices of $M$ is exactly $M$. This is the first definition.

Now it is always the case that $G|_{V(M)} \supseteq M$, so $M$ is an induced matching if $G|_{V(M)}$ contains no other matching, that is, $G$ contains no edge connecting vertices in two different edges of $M$. This is exactly the second definition.

I have never heard of mim-width, so it is safe to assume that it will be defined in any paper which uses it. However, it probably refers to the size (probably number of edges) of the largest induced matching in the graph.