Should I use oil or polyurethane to protect new cane chairs

What you will want to do is to get a polyurethane stripper with brush, a pair of rubber gloves, a scraping tool, fine grit sandpaper (200), a clean cloth, and a well ventilated room.

Start by applying the polyurethane stripper liberally and evenly over the wood using the brush. Let it stand for about 5 minutes or whatever the directions instruct you to do.

Using the scraping tool, gently scrape off the polyurethane in the direction of the wood grain, taking care to remove the scrapings as you go along using a clean cloth.

Once the scraping is completed, lightly sand the wood in the direction of the grains until the surface is smooth.

You're now ready to apply the finish of your choice, whether it be polyurethane, a stain, or paint. :)

For starters, I am going to guess you used a water based urethane instead of an oil based product? I have never seen a good oil based product react as you described to simple spills. I have seen some damage caused by very hot items being placed on a urethane finish, but normally, liquids will bead up and not penetrate the finish. Even though the water based urethanes are easier to work with, it take two to three coats to give the same protection as one coat of oil based urethane.

More directly to your question, the hardest finishes you could use as a DIYer would be a good oil urethane or lacquer. Lacquer finishes are common on high end manufactured pieces, but really need to be sprayed on, several coats to make a good protective finish. The end result of a lacquer finish is much like the finish on a car, very hard and durable. I have built many bars for folks and used 3 to 10 coats of oil based urethane with great results. I have never had a complaint similar to the problem you are experiencing. If I were going to fix your table, i'd probably start from scratch. Sand it down again, fix any color flaws then apply 4 to 6 thin coats of oil urethane, sanding with 220 or 440 between coats. I personally prefer using a 4 or 6 inch good quality foam brush instead of a bristle brush. Use thin coats, resist the temptation to put on too much at a time. Each coat applied and finished with long complete strokes across the entire table. Be sure to allow proper recoat time as prescribed by the manufacturer. Don't rush it! Not allowing proper dry time will create a "softer" finish as undercoats will not have gassed off enough and not hardened or cured.