Car won't start in cold temperatures

Let me clarify. The car always turns over, sometimes it starts for 1 second and dies, and sometimes it doesn’t start at all, but just turns over. So the engine will run for a split second and then just fade off and die, which the mechanic described as “starving for fuel”. But apparently the fuel system was in working order.

When I wrote about turning the engine on for the “cold start”, I actually mean turning the ignition key to ON to see if the engine will start. I didn’t realize that this was not supposed to be done in fuel injected cars, as it came directly from the owner’s manual. The owners manual has a section for “cold start”(hold the gas pedal down and turn the ignition to ON for 5-10 seconds; then try and start the engine normally as you would), and talks about how this gets a too rich of fuel out, but was confused by this. This “cold start” process, which to my surprise, actually has worked on a few occassions this winter and previous winters, but after many repeated “cold start” tries. Unfortunately it was a no go this time.
I really don’t want to have to rely on a “cold start” the next time it happens, especially if this is a no-no on fuel injected cars.