ginge, you are partly correct (for NZ)
Under 125 all training captains require an A, B or D cat:
125.567 Flight crew member instructor qualifications
Each holder of an air operator certificate shall ensure that any person
carrying out functions as an instructor in its flight crew member training
programme established under this Part—
(1) has satisfactorily completed the training required by this Subpart
to serve as pilot-in-command in operations; and
(2) holds a Category A, B, or D, flight instructor rating; and
(3) completes initial and recurrent training requirements applicable
to the instruction carried out.
Under 121 it is a different qualification all together.