I see your point . Where I think TC is wrong is that EOL is engine off landing . If you do an auto , it is an auto regardless of the engine state . You can just dump the lever with full engine and it is still in autorotation in my view ( aerodynamic ) .
I also agree that v few people demonstrate actual engine off landings partly I expect due to cooling off period but mainly I guess as it would be an added risk for v little gain !!
No doubt God will tell us otherwise.... a bit like USA v UK with settling / LTE etc etc I think we have got used to calling engine in idle an EOL .
Is it worth arguing about ?!! I rather foolishly tried to tell God that EOL is not relevant to the autorotation on the way down , where you have the option to keep the engine if you wish ( if it is still producing some power ) . An auto is an auto regardless!! End of !