Induction in transformer coil relationship to magnetic field lines in core
From my understanding is the process of induction described by a
  magnetic field line which approaches a wire perpendicular


That is the wrong idea. This is the right idea: -




  Does one need to differentiate between induction by magentic field
  lines cutting a wire perpendicular and the sum of magentic field lines
  inside a loop of wire?


Induction is not due to lines of flux cutting the wire perpendicularly. If you are talking about this scenario: -



Then it is incidental that some magnetic flux lines are passing through the conductor.