How to Determine if USB mouse supports PS/2?

Unless it comes with PS/2 support clearly noted, its best to assume it doesn't. As noted on this Wikipedia page, those adapters are nonstandard, so even if the mouse supports PS/2 you might be looking at the wrong adapter. Also, anytime you adapt back to PS/2 you're going to lose features of the mouse (the high DPI settings in this case).

If you really need PS/2, an active adapter should be able to handle any USB mouse - but at their cost, you're better off using a USB mouse (even if that means adding USB to your PC).

Active USB to PS2 convertors should just work. We used a few of these in servers where old servers still needed PS2 input and we only had a KVM with a 'USB mouse' output.

On the other had, you did not link to such a device. The device below is a passive plug which does not convert any signals. It assume that the mouse can do both USB and PS2 protocols, and that the mouse will detect which one it needs.

This means that you can not use this on a mouse which only understands USB.

Then there is a minor matter of the purple device you linked to. Colours from that era usually use orange or purple for keyboards, and green for mice. I am not sure that a keyboard plug is wired the same way as the green mouse plug. (The website you linked to call the one a replacement USB keyboard to PS/2).