I don't really know much about the swap tricks for FAT PS2s (I own a slimline, and only recently got a FAT)...
I placed the Need for Speed disk in, got to the main menu, press the eject button, swapped discs, pressed eject button, went to Online and then I got that red screen that asks to insert a valid PS2 game. (Did this at least 15 times)
You're supposed to open the PS2's drive to swap your discs without your console knowing right?
I remember that FAT PS2s had a flip top, a special drive circuit board that intercepted the eject button's signal and the infamous knife trick as methods for the user to swap discs without the PS2 knowing.
SCPH-10000 S. MINOKAMO v1.01 (Defunct)
SCPH-10000 S. KISARAZU v1.00 (faulty)
SCPH-15000 S. KOHDA (With warranty seal!) :D
DESR-5100 S. EMCS