In my day guys we were taught basic "T scan" when doing an instrument take off and climbout.
The basic scan was the" radial scan" with the AI at the centre (hub and spoke basically) and then a "selective radial scan "for specific manoeuvres where you use the AI as the hub and scan other instruments such as say the altimeter and HSI during a level turn more than the airspeed, VSI and Turn coord.
I assume White and Fluffy that is what that second instructor you spoke about is teaching???
AS arm out the window says dont stress too much, use what they teach in the beginning and you will feel more comfortable as your experience grows and can then use a combination of methods if you find it works best for you.
I have had a CIR for over 20yrs and now I just "Use the Force", it works for me but then again I went to "the Dark Side" years ago.
Remember that single pilot IFR esp in a light twin that may not be too well equipped and like most of them have marginal performance esp on 1 engine is one of the most difficult and demanding flying tasks esp when you are low IFR AND low Twin time.
So fly cautiously and well within your limits (and the aircrafts) and learn something every trip and you should be just fine.
Remember ANC,...Aviate , Navigate, Communicate. in that priority order ALWAYS!!!
It CAN AND WILL DEFINITELY SAVE YOUR A** esp when you are alone, tired and it all seems to be going pear shaped.
During VFR ops it is very important, during IFR ops in IMC it is VITAL!!!
It has saved mine.
Best of luck with your training.