Lines 1 and 2 both work on Phone #1 in Jack #1 (using 3-jack device w/L1, L2 and L1+2).
Lines 1 and 2 work both work on Phone #2 when it trades places with Phone #1 and uses Jack #1(using same 3-jack device).
Line 1 only, but not Line 2, works on Phone #2 when it uses Jack #2 (with either 3-jack device above, or 2-jack device)
However, I tested Phone #1 on Jack #2, using 3-jack device, and both lines work just fine. This is driving me crazy. Have I tested for all the variables? What could be causing this problem? Both phones seem to work, and both jacks seem to work. Things only fall apart when Phone # 2 is connected to Jack # 2. Why would this phone and this jack be incompatible?

Check that your line cord is 4 conductor, usually you can see it. If not Replace or test it.
Center 2 conductors is usually line 1(Red and green, or blue white/white/blue), out from the center 2 is yellow/black, line 2 or blue white/white blue for category rated wiring.