Not really correct to say that all injected engines use a mixture lean start. Depends on the manufacturer. As a rule of thumb Lycoming's injected opposed engines use a lean start whereas Continental's equivalent use a rich start.
I get the impression that Lycoming's injection priming leaves the engine a bit too rich so cranking with ICO eventually leans to a combustible mixture. Continental chose the opposite. Is this really correct? Damned if I know but that seems to be what's happening.