Mystery solved, no thanks to the official Toyota Parts Diagrams (via partsouq). If you look through the two sections of the parts diagram containing the Vane Pump, there's one section where the "Seat" is mentioned. However, it's in the wrong spot clearly as it shows it integrated into the fitting on the low-pressure part of the Vane Pump. Also, there is no "Seat" on the Hi-Pressure assembly either, where I was pretty sure this part belonged:
Then looking at the other diagram, focusing on the Power Steering hoses, there is no mention of an external Seat on the Hi-Pressure hose connection to the Vane Pump. I figured if it were anywhere, it would be here since it's technically not a part of the Vane Pump, but floats between the hose and the pump. Still, no dice:
Last ditch effort was to check the FSM. And as always, this proves my single most valuable investment outside of my stubby Milwaukee impact wrench. SR-46 clearly shows the seat as being in between the high-pressure hose and the high-pressure connection on the Power Steering Pump (Vane Pump):
Furthermore, there is actually a section in SR-49 that talks about replacing this part, called the "Union Seat," language that is never seen in the official parts diagrams anywhere. What's interesting here, is that it seems from the factory, these Union Seats were actually screwed into the Power Steering pump, requiring an extractor to remove. However, if one were to replace it, they are only available in floating style (the kind I obviously have as it fell right out and was not threaded):
So this is the first time I haven't been able to find a part on the official Toyota Parts Diagrams. And I'm still not sure if the "Seat" mentioned in those diagrams, incorrectly placed on the low-pressure return fitting, is the right part. Luckily, I still haven't lost this part even though I removed it almost a month ago (checked last night). Therefore, no need to order a new one.
Hopefully this helps others who may find this part mysteriously appear after disassembling the PS system. Just go to the FSM first, I need to follow that advice from now on.