When can I log PIC time in a new category of aircraft?
Short answer: you can log ASEL PIC time only while flying solo, not while flying with an instructor.

You seem to be mixing up a few regulations: you quoted 61.63 but the text is actually from 61.31. However, since neither of those regulations are about logging time anyway, they aren't directly relevant.

The rules on logging PIC time are in 61.51:


  (e) Logging pilot-in-command flight time. (1) A sport, recreational,
  private, commercial, or airline transport pilot may log pilot in
  command flight time for flights-
  
  (i) When the pilot is the sole manipulator of the controls of an
  aircraft for which the pilot is rated, or has sport pilot privileges
  for that category and class of aircraft, if the aircraft class rating
  is appropriate;
  
  (ii) When the pilot is the sole occupant in the aircraft;
  
  [...]


(e)(i) says that you can log PIC time as sole manipulator only if you're rated in the aircraft. In your case, you aren't rated for ASEL (yet!) so you can't log ASEL PIC time while flying with an instructor, even if only you touch the controls. This is probably what the school is referring to.

However, (e)(ii) says that a private pilot who is the only person on board the aircraft may log PIC time, without any additional requirements (although other regulations do require an endorsement in order to solo, as you said). 

That means, you can log solo ASEL time as PIC, but not training time with an instructor.