OK, here's a counting method that works for all the standard systems below, although it calls 120/208V 3-wire 2 phase (which is plausible in my book, although it's a system only ever used for single phase loads).
Namely, write down all pairwise voltages between conductors. For the voltage with the highest count (up to small differences, and choosing the larger voltage in the case of ties, to be unambiguous), choose any pair to be phase angle zero, and plot the phase angles of all those pairs. The number of phases is the number of lines through the origin spanned by those vectors. I.e. the number of different phase angles up to sign.
2-wire: 1 phase
120/240V 3-wire: 1 phase
120/208V 3-wire: 2 phases
120/170V 3-wire: 2 phases
Delta 3-wire: 3 phases
Wye 4-wire: 3 phases
Hi Leg Delta 4-wire: 3 phases
2 phase 4-wire: 2 phases
2 phase 5-wire: 2 phases
Cheers, Wayne