From AIP Definitions: as you are already aware-
Medical Flight: A flight providing transport of medical patients, personnel, and/or equipment, prioritised as follows:
MED 1: An aircraft proceeding to pick up, or carrying, a severely ill patient, or one for whom life support measures are being provided.
MED 2: An aircraft proceeding to pick up medical personnel and/or equipment urgently required for the transport of a MED 1 patient, or returning urgently required medical personnel and/or equipment at the termination of a MED 1 flight.
see
AIP ENR 1.10 Flight Planning, Appendix 2 (top of AIP page 614) it says
STS/ for special aircraft handling; eg, STS/MED 1, STS/MIL SPEC REQ.
MED 1 etc. is an entry on the flight plan to notify ATC as quoted above.
It is the reason for the flight. (just as an RPT flight remains an RPT flight even if somebody gets sick and requires urgent medical attention). On an RPT flight with an ill pax/crew on board, a decision is made as to the actions required to safeguard that persons well-being as much as is possible within operational limitations. If that decision requires ATC to provide the aircraft with additional services or priorities then the appropriate method of advising of those requirements is with a PAN call. You then bypass the queue for priority handling.
It is a formality to advise ATC of your need for special handling.
regards