ok, i don't know if this is the same problem, but my friend just bought a sled from a guy who said it did the same thing. it would not start when cold with the electric start, but after driving ten/twenty minutes, it would work just fine.
we took the starter apart and i think one of the little brushes was gone, and it just looked like it was in rough shape. he is getting a repair kit that will hopefully fix it.
1992 Polaris RXL 650 EFI 1990 Skidoo Formula MX 470 1997 Skidoo Grand Touring 500 1997 Arctic Cat Pantera 580 1979 Ford F150 ~360 5.9L, Edelbrock intake, 4bbl carb, 4 speed, divorced transfer case, 3" body lift, 3" suspension lift, 33's, redone interior. 2003 Dodge 3500 srw 6spd
I believe there is a throw spring that makes the starter engage once it starts spinning.
moisture usually results in a dead spot where the starter won't spin at all but its worth having a look at especially if thats when the problem started