I'm sorry to report that you'll be unable to use the PS2 USB ports for recording purposes. The main reason is that you need drivers for the PS2 for it to recognize your video recorder.
HOWEVER, the traditional method of connecting your PS2 to your video recorder through the A/V cables (& your video recorder to your TV, so you can see what you're doing) is guaranteed to work as it's normally used by world record video game players to record their submissions on VHS.