You might try using a razor blade to gently scape...not cut...scrape with the blade flat on the table and try scraping the ring out of the finish, then respray it with a finish coat, this might work if it's only in the finish, if it's through the finish this won't work

This cloth is made especially for removing (white) water stains, it's what I use and have very good results with it.

Not sure what the "darker" ones are. They may be from moisture getting into the wood (and not just into the finish). The only cure for those is stripping and refinishing.

Here is a link that might be useful: Stain removal cloth

If it was from a hot pot of some sort. That ring has penetrated the finish and has created moisture under the finish , that's the ring you see.
There is no easy way to remove the ring, short of refinishing.

Kmealy, I'm curious about that "cloth". It looks like nothing more than a microfiber cloth ? Is there a "product" on the cloth ?

Yes, it has tallow acids and ultrafine silica. It will leave behind a little grease that's easily cleaned off. If the resulting sheen is too high, it's easily dulled back with a little aerosol.

Oddly enough a similar cloth is sold to polish brass band instruments (trumpets, trombones, etc.). I used one last week to get a scuff out of a hammered copper table top.

It's also good at removing silicone oil.

They usually carry a couple of brands of these at Lowe's, though some are in a plastic container that you can't really tell what they are.