After reading books before lectures, here's my thought regarding inner models and so on. Correct me if I am wrong.
So there's universe $V$, which we assume to be the true universe. By Gödel's incompleteness theorem (and the need to assume large cardinals etc.), we cannot prove that inner models that are subclass of the universe exist, but if they do, then the inner model satisfies the ZF(C) theory (axioms). Standard model is an inner model that uses actual element relation $\in$ in $V$ - we cannot prove the existence of it again by Gödel's incompleteness theorem. Sometimes inner models refer to standard models. Standard models may be both internally and externally well-founded but may not be transitive - in such case, Mostowski collapse lemma guarantees that there exists a unique transitive class model. If the model is not well-founded externally, then $V$ would see it as not well-founded so the talks about Mostowksi collapse lemma are of no use. However, as $V$ is assumed, we may instead assume that this model is universe $V$.
If standard model exists, there is a minimal countable model that internally satisfies ZF(C) theory, but this model is externally not well-founded, while standard model is.
Is this correct understanding?