Another quick update:

Qsilver7, I found your original post. I switched left and right (identical) purple relays, no joy. Checked fuses 82, 84 - all good.
Took apart the trim and got to the lock. My lock did look like the one in dbx5's thread but turning the star screw did nothing for me.
The exterior button, previously not working, worked for a minute while I was tinkering with it but now not working anymore. There are no loose contacts anywhere that I could find.

Right now the lid will open only with remote or button on dash. When I shut the lid it there is no sound from the lock and the door is not being grabbed.
Today it opened again while I was standing at a light! luckily I was quick to jump out and get it shut before the light turned green.

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2004 x5 3.0i Silver
MFGDT 2003-12-11