Jap PS2 was bought in Hong Kong (but can be bought at Computer Exchange, Rathbone Place London). Came complete with UK mains adapter (because HK is UK mains compatible). The Boot Disk is actually a Utility Disk that enables drivers to be loaded into the memory card.
Used with a modchip.com mod chip, the boot disk gets the system booted and via the swap trick, the backup copy is inserted. We could just as well use any Jap Original Game if we had one (which we do).
I'm not sure if the current modchip version can use this boot disk and indeed I didn't upgrade my modchip to their latest version when it was sent to me. (www.modchip.com). No probs to date.
With a PAL TV and AV leads I play NTSC backups with a NTSC-PAL convertor in place; I play PAL backups with that device removed. US PS2 will require the same treatment I reckon.

Good luck