First, get the fuel pressures checked. Then have the injectors tested. It sounds like the cold start is working fine. That's why the car fires up when you first try to start it. The cold start only operates for a second or so through the thermo-time switch(unless it's really cold out). The further attempt to start is without the cold start valve(since the switch has heated up from the current flowing through it). Either the system rest pressure is dropping or(more likely)one or more injectors are leaking down. One(or both)of these should be it. I wouldn't go after the cold start injector for the reason mentioned above. And failures are really rare.
Without testing first, anything you do is going to be guess.