Wow. I can't believe that is even arguable.
ALWAYS start and stop the engine at low rpm.
Upon start up give it a minute or two to warm up. Same for stopping. You want to lower the rpms and let the engine cool down before killing it.
That's what I do.
My Dad and common sense always told me to start slow to warm up and shut down slow to cool down. I have never done anything different on any engine.
I always start as slow as possible and let engine warm up at least 30 sec to a minute at idle and then gradually advance to full throttle over the next 30 to 45 seconds. On shut down I have always gradually advanced to idle speed and let idle for 30-45 seconds before shut down.
However, the dealer I just purchased a new Grasshopper from (Vanguard 29hp engine) told me to always shut down at full throttle. When I questioned him he said air cooled engines cool best at full throttle and will overheat at slow speeds. Did not mention starting speed.
My Vanguard engine manual says ---
To Start -- Choke and move throttle to fast and start.
To Stop engine -- Move throttle to Slow and turn key off.
I plan to continue my old ways and follow Southern Pride and my Dad.