AirMed International is an actual Part 135 Air Carrier that owns it's own aircraft and conducts patient flights. You are getting coverage from the folks who actually do the flying and patient care. They do both CONUS and international operations. They are also the main DoD contractor for this type of service.
https://airmed.com/About-Us.aspx
"With state-of-the-art medical jets and the most experienced and respected medical teams in the sky, AirMed is able to deliver on its mission of an uncompromised standard of care. AirMed owns and operates all aircraft, as well as employing its own dispatch, flight, and medical teams. This vertical integration ensures a seamless service and world-class care 24/7/365."
AirMed provides worldwide services from multiple bases strategically located across the globe. With accreditation from both CAMTS and EURAMI, AirMed is able to deliver a seamless transport experience to our customers regardless of location.
MedJet is a broker and they don't actually own or operate their own aircraft so they have a "network" of part 135 carriers that they call upon.
"[...] Medjet is an authorized Indirect Air Carrier (IAC) utilizing the services of licensed Part 135 air carriers to meet your air ambulance transportation needs. Medjet does not own, lease or operate any aircraft, and has no affiliation with any direct air carrier. As an IAC, Medjet contracts for the provision of air transportation services in its own name and coordinates the provision of medical services for your flight. All flights are operated by licensed direct air carriers."