You said you shovelled snow the day you ran on idle?
When you get steam in those conditions it's usually a sign of coolant freeze up and bad news. There should be no problem idling these bikes for 10 minutes OCCASIONALLY - the dealers do it all the time for diagnostics and to test the fans! However the fans will be running after about 5 minutes and usually when the temperature guage hits 2/3rds. Were both your fans running at all? Fans should come on, before any warning is given, Can you turn the blades easy with your fingers through the grills? If you've been to scared to idle the bike on the stand, you risk having fan failure without knowing it until you are far away in city traffic in Summer.
The temperature guage, light and fans seem inconsistent, which suggests not all parts of the cooling system are at the same temperature - which they should be if the system is thermally stable.
Assuming your fans are ok or you know they are working (do an electrical check first if you are not sure. Also check for any blown fuses: siezed fans usually blow a fuse) You have probably got air in the system and it can be difficult to get out (check the archive forum) but keep in the back of your mind if you had fan failure that's good news and probably responsible for the air lock. If it was a coolant freeze up, - something else you won't wish to contemplate and may have nightmares about. Often you may ride miles not needing fans, then the first time you idle and need them, discover they are siezed and not working.
There are posts in the archives which recommend 'burping' the hoses at the front to get air out. On these bikes the reservoir tank is only just above the pressure cap level. I read some bikes had caps on both radiators which makes the air purge job easier, my K has a cap on 1 rad only. If your cooling system is stable on idle, you should feel the same temperature evenly across the surfaces of BOTH rads. As the bike warms up (slowly on the stand) The bottom hoses will get warmer as water circulates. If the rad tops get really hot and the bottoms are stone cold, look for a thermostat or pump problem.
You need to be careful because your temperature guages and warnings may not tell the truth in a fault/vapour lock situation. Experienced hands can usually guess an overheat motor without a thermometer or guage. The bike sensors only measure when coolant is in contact and may show a low reading if there is an air/vapour lock around them.
Assuming a good thermostat, you can usually remove air and check if the sytem is stable by removing the header cap, which drains the reservoir, and running the motor at idle a few minutes whilst 'burping the coolant pipes. If the temperature guage climbs slowly until fans comes on and the rads are both evenly warm with no other warnings you have a potentially stable sytem that might be ok under pressure. If you get erratic temperature readings or coolant shooting out of the open radiator after 'burping'- you have a more serious problem. If everything seems stable and ok, replace the pressure cap. There are suggestions in the archives to fill the reservoir, allow the motor to cool (which pulls coolant back into the system) then top off reservoir to mark, run bike and check system temperatures stay stable running on the stand until both fans come on - don't push more than 2500 rpm though.
Most pressurised engine cooling systems can be run without caps in emergency or for short test periods unless you're pushing lost of power through a motor on a hot day. If you ever get a hose leak or a small hole in a rad, depressurising is the best get you home method.
Don't run the bike on the road with an unstable cooling system and remember in this unstable state you cannot be sure the guages are telling you the truth about engine core temperature. Hope some of this is helful.
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