I'm reading through Joel David Hamkins' set theory lecture notes. On page 14, on the subject of inaccessible cardinals and submodels of ZFC in $V$, he defines a universe cardinal to be a cardinal $\gamma$ such that $V_\gamma$ models ZFC. For example, if an inaccessible cardinal $\kappa$ exists, then $\kappa$ is a universe cardinal. But furthermore, by the Löwenheim-Skolem theorem, there are then lots of models of ZFC inside $V_\kappa$, and Hamkins says that this proves that $\kappa$ is the $\kappa$th universe cardinal. Specifically, $\{\gamma < \kappa: V_\gamma \prec V_\kappa\}$ is evidently closed in $\kappa$, and is "unbounded by Löwenheim-Skolem".
This is confusing to me. The Löwenheim-Skolem theorem says that $V_\kappa$ contains submodels of arbitrary cardinality. But in order to have any universe cardinals at all, you need models of ZFC of the form $V_\gamma$, which don't seem to come from Löwenheim-Skolem. Why should these exist? Is there something you can do to a model to put it in the form $V_\gamma$?
Relatedly, it's also true (this is in Kanamori's Higher Infinite) that the only models for second-order ZFC are $V_\kappa$ for $\kappa$ inaccessible. Can someone explain the difference between these two concepts? I'm finding the distinction between first-order and second-order Replacement a little hair-splitting.
I hope I've phrased this the right way -- I'm a set theory rookie. Thanks in advance!