You've checked resistance across the different coils? Did you find anything there?
Did the failed coil short out or break continuity?
Gauss' stuff always confuses me for some reason, I had to take e-mag twice back in college. I know that core diameter is part of the equations in working out what's going on in a system like this. In a reversed situation, buzzer, the smaller core will energize and release quicker but be weaker at the same voltage.
And I'm probably half right and speaking to someone who actually knows way more about such things, damn, forgot for a moment what you do outside of mopeds. Heh.