[blue62]

Sorry I was thinking you had a cold start problem.
Every aspect of modern day engine operation and management is via electrical signal. You on your 2002 Boxster and I on my 2000 S don't give the engine fuel or even air. We just push a pedal that is connected to a potentiometer and the ECU does everything else.
So yes a failing voltage regulator could be the root of your problem. or at least a contributing factor.

So I would replace the voltage regulator first as it is a known problem.

If you still have issues.

I would still do a vacuum test with a proper vacuum gauge as that simple test can give you a lot of info on general engine condition. Vacuum leaks can cause a lot of tail chasing.

Also when you replaced you plugs did you check the condition of the coils??? No cracks??

Then:
I would do a fuel pressure and volume test next.

If you have disconnected your battery for any reason you need to recalibrate your throttle. (gas pedal).
If I remember correctly you reconnect the battery.
Turn the key on for 1 full minute ( do not start engine: do not touch the gas pedal).
Turn the key off for at least 10 seconds then start the car and let the computer re-adapt.