"Medical Reserve Corps"? An acceptable moniker.
"Ham Strike Team"? It conjures up whackers with badges, emergency lights and sirens, HTs worn everywhere and always on, more antennas on their POV than the local hospital roof, and always ready to respond - to what, no one knows and with no defined and commonly understood tasking. What service would they provide in a critical incident? Improvised installation of a 70cm Yagi in a tornado?
Well, one thing that they have provided was directing the people that went to the civic center for the mass vaccine event so the DFR people could handle just the inside part (registration and directing to the medical teams doing the vaccines) so only few city crews were required for the event while the hams and other volunteers handled the line of cars, parking, and directing the folks into the building (at the required minimum distance between people) and kept the folks outside of their cars to a small number to help prevent transmission of the virus.
Please note that in this event, no radios were allowed or used by the outside folks who used colored flags to communicate. This was at the direction of the event organizers so no illegal radio operation was done. No HTs. No emergency lights or sirens. No antennas. Well, the DFR folks did have their HTs and their ambulances, engines, and trucks did have their radios, emergency lights and sirens, and antennas but they were all parked in a secure area with the equipment shut down. Now there were badges, but those were provided by the group running the event and this was to let the officers know that they were vetted and supposed to be in the restricted areas and directing folks along the way.
I'm not really sure that volunteers with colored flags would count as a "whacker". Yes, "Ham Strike Team" probably wasn't the best term, but it was what the organization used so what can be done. Personally, I might have suggested "Volunteer Communicators" or the like. I guess that's what happens when it's not the hams that work on the license application but folks that don't understand radio operations (after all, it's not using the radio that's the point, it's providing medical services and the radio is just a tool like any other) and all they know is generally it's a group of hams helping them with communications. At least they did get some help (probably from the city's radio department) on preparing the technical side of the application.