In-Place Insertion Sort
Naming index and nextIndex as i and j would have been OK.

There are no comments, which is not necessarily a problem. But a one-line javadoc for the function would be helpful. I know, I know, from the signature it's kind of obvious, but still. For example, what if caller passes in null? Well, we throw a fatal exception, and that seems appropriate, caller gets what he deserves, he violated the contract spelled out in the javadoc. But only if there is a javadoc explaining the caller's responsibility.

The inner while loop would be more naturally expressed as a for loop.

The inner while loop might become a helper function. Then you'd have the opportunity to give it an informative name, perhaps copy.

That said, overall the function as written is reasonably clear.