Every reputable organization, including most gun control organizations, sets the criteria to be:
One shooter
3+ or 4+ killed, not including the shooter
Not related to an act of terrorism or a state of war
Unrelated to any other criminal activity
This has been the criteria for over twenty years, and is the international standard. The only groups that are pushing for the loose definition are those that want to give political candidates a big scary number to proclaim.
Yeah, it's still sketchy to hand-wave away all those shootings because of a definition. That as you wrote it is specifically the FBI definition of mass-murder. They don't have any definition of the term mass shooting whatsover. Sometimes some reports have used it a synonym and others have not. But the point is if you're pulling a "definitions matter" thing, then the difference in wording also matters. Is a "shooting" and a "murder" exactly the same thing now? It seems a bit rich to label politicians liars because they defined "shooting" as "someone getting shot".
If we say that 273 shooting incidents happened in a year where 4+ people got shot per incident, and we refer to that as a "mass shooting" then it's not actually making it any less by saying that it's wrong to call them "mass shootings" therefore the topic doesn't deserve further discussion. Those incidents still happened, according to the criteria given.
Also, it makes little sense to whittle that down further by saying if they were gang members or it was crime-related they don't count either. All those things are part and parcel of America's gun problem.
Why would it make any sense that when talking about gun legislation you
don't include crime-related stuff in there? So it only counts as gun crime now if someone loses it for completely baffling reasons or something.
So it's easy to whittle down the definition as much as possible, say that only 2 people a year count as "mass shooters" then point out how such-and-such legislation wouldn't have stopped those specific two people getting guns, so the legislation is useless. But that's only because we narrowed the definition so much that it missed all the actual effects of the legislation.