As I said, I was playing Devil's Advocate; actually, I've been in both scenarios and my take is you comply with the AMP (a). Depending on the type of aircraft, the AMP includes TC, ATA 5, MPD,MRBR, Schedule or whatever recommendations/requirements but the operators can and do add their own. It is approved by the regulator and you comply. The term "overhaul" does not appear, for example, in the 747-400 MPD; the term is "restore" but that includes work up to a full overhaul.
In 2002/3, while auditing and on-site with the conversion of the BA 757s to SF for DHL, the AMP - written by Boeing for DHL - stated the case I mentioned and off came the actuator because a repaired item was fitted during the concurrent "C" check and an "overhauled" item was stipulated.
Component requirements are stipulated in the AMP. I've just been involved in adding carb overhaul stipulations to a BN2 AMP based on an FAA SAIB (match carb overhaul lives with those of the engine) that the UK CAA insisted was included.