If the battery is getting weak, terminals corroded, cables frayed.... Then when cranking the voltage may be reduced enough to screw up the computer, injectors or fail to generate spark. Push starting it could unload the electrical system enough that other stuff still works. The battery is certainly weaker when it's cold, so if some of the other things listed are a problem, they are aggravated when it's cold.
Stick a voltmeter on the 12V at the cigarette lighter for instance and see what it does when you crank the engine. Once you have run the engine for a few seconds the cranking load is much less than right out of the box. I'll bet you can see the difference between the first cold morning start and the second one, no matter how you got it started.
