Look in the water heater compartment and find the model number of the water heater. If it's one that has electronic ignition AND propane/120v operating modes, my guess would be that the black switch turns on the propane side and the white switch turns on the 120v side. That would be consistent both with the types of switches and the behavior of the indicator lights you described.
Il Carriaggio — 1975 Argosy 24 | Il Progetto — 1976 Argosy 28 Center Bath | WBCCI# 15566
He has all of the virtues I dislike and none of the vices I admire. — Sir Winston Churchill