when u push start the car, is the engine RPM higher than the RPM on key start? and if it is how much? how long do u need to push it?
THE ONLY DIFFERENCE BETWEEN PUSH START AND KEY START in "starting" way is engine RPM, there is nothing else that can do it start better, than why the engine needs higher RPM to start can be compression or pre glow fault, check valve clearence and that the PRE GLOWING works OK, also check so the fuel dont leak back to fuel tank.
hope u understood my english it aint the best u can find... =)