[Mikendebbie]

Our table looks like it has a thin wood veneer top - not a plastic laminate top.
DW has a "white ring" remover that we bought at a Ace or True Value hardware (see pic below). I think this has been around a long time because Debbie remembers her Mom using it when she was a kid. The included rag has some chemical treatment in it that will effectively "sand" or buff the white ring out. "Sand" is my word for it...but it works great. You have to put a lot of elbow grease into it. I have rubbed and rubbed on some real bad water rings for maybe 30 minutes - but the end result is great. You might try this stuff and see if you can get similar results. This little bag lives in a drawer in the living room and gets lots of use after Thanksgiving and Christmas family visitors go home.