So on the second try, it starts and dies.

But exactly WHAT does it do on the first and third try?

The SKREEM issue goes like this:
.... Turn key, starter spins engine over, it starts and dies.
.... Turn key to off, then back on and to start, it starts and dies.
.... Turn key to off, then back on and to start, it starts and dies.
.... Turn key to off, then back on and to start, and NOTHING happens.

How does your situation differ?