Why does this USB device draw different currents from different sources?
I suspect the device you have will takes power from both the 5V line in the playstation graphics port and the usb input (possibly fed through a diode). The output from your wall adapter is likely slightly above 5V but the playstation will have a well regulated 5V supply. Because the voltage of the wall adapter is slightly higher that PS2, the device will draw most of its power from the wall adapter rather than the graphics port (or you may even be injecting powering into the PS2 via the graphics port), when you use the PS2 usb it draws most of its power from the graphics port.

You wonder why the adaptor needs so much power, firstly the adaptor has to do quite a lot of work - it takes the 3 analog video signals and digitizes them at something like 30MHz before sending digitized data out over HDMI. Secondly an HDMI source provides a 5V line for powering electronics downstream, and your adaptor has to provide that power as well. It sounds like the manufacturer has been conservative and states that you need at least an amp to cover both things.
If the converter has been designed to run from the PS2 then I would do that.
The dangers are:

a) using the PS2 sockets, that you overload the power supply/sockets in the PS2

b) 
or using the power adaptor, that whoever designed the device never expected it to be powered when the PS2 wasn't and didn't take any steps to prevent it feeding power back into the PS2, or forcing whatever drives the video output into SCR latchup.