Is there a meaningful difference between institutions that are extractive and institutions that rent seeking? Is battling one battling the other?

They’re quite old terms but they have very specific definitions. A rent seeking institution is one which extracts resources from other people through an asset without creating incremental value for those other people in the process, generally through social or political influence. It was originally used to refer to landowners who get paid just for owning the title to the land, and not themselves making any improvements or developments to that land.

An extractive institution is one which depletes the resources of other people, actively contributing to the decline of the wealth of those people.

Generally, being extractive is worse than being rent seeking. Rent seeking doesn’t usually reduce the amount of resources to go around and can even be sustainable over the long run.