I'm in love with this rug and can't find it anywhere. Please help! Thanks!

Your last image (inspiration?) didn't post, but I think I know what you're going for and I approve. A question: are you using furniture you have or purchasing new for this? A classic round game table might be better if you aren't repurposing an existing table. Or an expandable 5' round dining table so you CAN make it bigger if needed. Having to fit existing furniture in is of course a different job than picking out new stuff!

Hmmm, did you take the photo in a store, or a friends house, or.....? There may be a label that shows the brand and the name on the back side, if you could 'visit' that one again. :-)

Do NOT use barkeepers friend....unless you want to etch the whole table top. If the rings are not visible from the top, I would ignore them...likely there will be more and eventually all will blend together. The thing to remember is you can't "un-etch" something....you need to polish it all down.