do as your checker says, for much easier progression
Whilst I doubt anyone could find cause to argue with that statement, isn't there room for the captain's judgement to override? Why chop the engine during low level manoeuvring to redirect your climb away from steeply rising terrain and add to the workload of both of you in a situation where margin for error is much less than you're used to? Making use of that thrust available under steady state conditions seems to make sense to me in that situation.
On the other hand why would you be content to wait on completing the acceleration when climbing straight ahead over flat countryside? I'd rather get the fire out please.
the (correct) FAA procedure
I thought the longer you worked in this game, the more you realised it's mainly just shades of grey and good judgement prevails? If we only had to learn the correct procedures we'd be all be aces in no time. Both procedures work, they just optimise different criteria.