It's a very long time since I flew a B206 so don't speak with any authority here. If I recall the at idle wait time after start which you refer to is from a cold start, meaning the engine has been shut down for more than 15 (or is it 20?) minutes. If the engine has been shut down for less than 15/20mins you can go to full noise after start without having to wait at idle for any period.
The cool down time before shut-down also has to do with stabilizing the temperature of the turbine discs/blades. The blade tips cool down (and heat up) faster than the blade roots/disc, and allowing the temps to stabilize for a minute before shut-down minimizes the shock to the metal brought about by the rush of cold air into the engine on shut-down.
The cool down times vary of course depending on the engine, for PT6 I use 1 minute, Arriel 1S1/2S1 is 30 seconds, or whatever....