The '02 manual does not spec an idle speed in gear, but in park or neutral, it wants to see 750 +/- 50 if the aircon's off, 825+ if it's on. Sounds like yours is working. Every auto I've driven idles a little lower in gear, so while I don't have a spec for that, I'm inclined to call it normal. Turning on electrical loads or working the steering loads the engine, so it makes sense it's dipping a little more under those conditions. The vibration/stumbling under load, while it's still pretty close to its target RPM, makes me suspect a mixture issue.
MrElCocko may be on the right track with the MAF sensor. Yours sounds a bit like how my '93 behaved when the MAF sensor connector was worn out. Sometimes it would idle low at lights, sometimes it would surge, sometimes it would stall completely unless I shifted to neutral or drove with both feet. Wiggling the connector would clear it up for a while. Finally I got around to replacing the connector, and haven't had a problem since. I don't know if the R50 suffers from the same issue, but I would check that the plug feels tight and the sensor is clean. If that checks out, check for cracks in the boot between the MAF and the throttle body, and check for vacuum leaks. Might be worth throwing some fuel system cleaner in the tank in case one or more injectors are a little gunked up. One or two cylinders firing poorly or not at all could explain why it's vibrating so much, though I wouldn't expect that to only act up at low RPM, and I would expect it to throw misfire codes if it was that bad.
Probably wouldn't hurt to clean the idle controls, maybe check for signs of the coolant leak those get, but again, it sounds like that system is doing its job. If your throttle body isn't the fussy electric one, maybe give that a clean, on the off chance.