So long as you intended to fly when the aircraft began rolling, it is legally a flight. You can log the time.
Presumably, the school charged you because:
1) It took the instructor's time.
2) It consumed fuel.
3) The time on the meter counts towards when the plane needs its next 100 hour check, oil change, and engine overhaul.
True story: Pilot lands his plane. Parks it in temporary parking, planning to leave in a few hours. He goes to a bar with some friends. He has too many drinks. Decides he can't fly as planned. But he won't leave his plane in temporary parking. No problem, a friend offers him a hangar.
He gets in the plane, starts it, and goes to taxi to the hangar. A combination of too much speed, poor control technique, and weird winds causes him to become partially airborne and smash into a hangar.
Fortunately, he has plenty of witnesses who corroborate his story that the flight was unintentional. The FAA has no jurisdiction, since the operation was not, legally, a flight. Nobody intended to fly.
I think state he was in argued that they could get him for a DUI, again, because it was not a flight. He was certainly over the limit for driving too, and the FAA could act on his DUI conviction if they wanted to.
If your instructor won't sign off on the time, ask him this: "If we had crashed into something while taxiing, do you think the FAA would agree that this was not a flight? What if we were a passenger plane and had an accident?"
Oh, and to reply to: "Flight Time: .....commences when the aircraft moves under its own power for the purpose of flight and ends when the aircraft comes to rest after landing. Say it with me... AFTER LANDING. So no take off means no landing, which means no flight time, regardless of your intent to fly. Many argue otherwise, but it's RIGHT THERE in the regs, clear as mud."
Every time an aircraft comes to rest, other than before its very first flight, it comes to rest after landing. Or are you seriously arguing that if you stop on the runway, you can't count taxiing in the flight time (since it has come to a stop after landing), but if you roll off the runway you can? That's a pretty nutty interpretation.
If you were correct, something else completely nutty would be the case. If you got into an airplane intending to fly, rolled the plane, and then hopped out, you could count flight time until you landed the plane the next day and stopped. After all, the flight time started, but didn't end until after the plane's next landing.
That's an obviously absurd interpretation and I doubt anyone but you believes it.