As long as the ring is white, it most likely is entirely, or almost entirely in the finish. A very old ring may penetrate into the wood, but in most cases where that happens there's likely to be a brown or black "edge" to the ring. The brown or black color is from mildew.

Any abrasive method such as toothpaste, oil and steel wool, etc that removes the ring will mostly remove the finish at least to the depth of the ring. That doesn't mean you shouldn't use an abrasive, but it does mean that you need to reapply something to re-seal the surface once the ring is gone.

If you can get a satisfactory color match, almost any "furniture polish" will give some degree of "re-finish." I'm somewhat partial to Old English but that's entirely because its about the easiest to find here, it's relatively cheap, and I have some almost at any time. (And since I've used it for the stain on much of my furniture, the dark stuff matches my stuff.) Any "oil containing" polish will eventually harden at least to some degree, if you can avoid re-ringing long enough, and you can build up the protection with periodic re-treatment - once you've achieved the color match.

For removing the ring, a method that sometimes works if the stain isn't too large or too old, is just to put a piece of paper towel or close-weave cloth over the ring and run a fairly hot iron over it. This sometimes will "re-crystalize" the existing finish so that it resumes its former color. (The actual effect is to drive out the water that's formed a "hydrate" with the finish material.) It does require a fairly hot (cotton or linen setting?) iron (usually) for this to work, so be careful not to char any wood.

There's not a lot of difference in which nut you use if you want to rub a nut on it, but if you don't want to buy a bunch of nuts you don't expect to use in short order you probably can get a variety of cooking/salad nut oils at the local food market if you look in the "exotics" section. I have walnut, pecan, and probably a couple of others somewhere in the pantry. Whether the oil, rubbed in, benefits the table you can use the rest for a bit of variety in your salad. Just don't wait until it "turns" on the shelf, and if it's something new be sparing about how much you sprinkle on the spinach until you're sure you like it.

The only really sure way to remove an old or deeply embedded ring is to remove the finish, and reapply new. If there's penetration into the wood, you probably will need to expose bare wood to bleach out any mildew. Most finishes don't support mildew growth, so if there's black it's into the wood.

If you can figure out a compatible finish, you may be able to abrade out the ring and refinish locally, just in the spot where the ring was. A couple of applications of an appropriately colored furniture polish may work for the restoration. The local finish you apply usually needn't exactly match the composition of the original as long as you get some surface protection down without an obvious "edge."

The only real way to tell if this "local fix" will be satisfactory to you is to try it and see. If a "spot refinish" doesn't blend in to your satisfaction, the only really sure way to restore truth and beauty might be a complete refinish of at least the top surface of the table, so that what's "visually contiguous" all looks the same.

If you need to go to a full top refinish, you may want to further your TAS by getting a decent light-weight belt sander. I'd recommend at least a 21" x 3" belt (the most popular size in my area) since the shorter ones are a little more prone to "dig in" and the larger ones tend to be heavy enough to be pretty clumsy. Practice on some scrap wood before hitting the table top though, and stick to relatively fine grit belts (80 to 120 grit, or so - 40 grit is frequently offered at retail places but can be very aggressive for your use).

John