How do I identify if a caulk is water-based or solvent-based?
No definitive general rule that I know of.  Back in the day all silicone caulks could not be cleaned up with water, nowadays there are water-cleanup silicon and siliconized caulks.  It always says on the tube if it is water cleanup.  Also, the base doesn't really define how it gets cleaned up either.  GE makes some 100% silicone based caulks that while wet, can be cleaned up with water thanks to some chemical soup they came up with.  

Generally, latex and acrylic based caulks should be water cleanup.  Generally, silicone, butyl, (poly)urethane based caulks will not be water clean up. But, as I mentioned, there are exceptions.  You have to read the manufacture's specs.  When it is water clean up, that is a selling point, and will be obvious on the tube.