Polyurethane finish over stained wood – can I combine brush and rub on techniques

When you apply multiple coats of urethane, any dust or small flaws will be amplified with each new coat. If you want the smoothest finish possible, then wait until it is dry, very lightly sand with very fine paper (220 or finer) to remove any dust bumps, wipe it clean with a slightly damp cloth or tact rag, let it dry or buff it dry with a dry cloth, then apply the next coat. There is no reason to use mineral spirits, actually I would advise you not to use mineral spirits.

I see no advantage to applying successive coats with, then against the grain. As long as you apply an even coat with a good quality brush or foam brush and don't overwork it, any brush marks should disappear. The finished product should be so smooth that you couldn't tell which way you brushed the urethane on. Also, if you are using a satin finish urethane, be sure to mix it well, stirred, not shaken. The dulling agents tend to fall to the bottom, so it is important to mix it well. No matter how many coats you apply, it should not become "glossy".

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.