I had a simular problem. Batt was fine, but when I tried to start it, the solenoid was heard to "click" on, but nothing from the starter. If their is enough power to engage the solenoid, there should be at least a slow turn-0ver from the starter. Maybe a bad starter ground? Batt at 12v not enough to turn it over, but enough from a boost (14.6v).
Anyways, I had my starter swaped out, and problem went away.