I have never been required to log running time in a tech log or maintenance release regardless of the country I was working nor have I every heard of it happening.
In regards to the R22 I believe there are calculations in regards to how much flight time has happened if you log engine time. Haven't flown one for a long time and never worked on one so possibly wrong on that. Will leave that one at that.
Every company I have worked for log air time for the aircraft. That's what the manufacturers ask for. Bear in mind most machines were designed more than 40 years ago when Flight time meant flight time.