I think the term has been used interchangeably at times but I agree with your definition of a noir style character, or at the very least a hero set in the early Golden Age. "Mystery men" evokes a certain image in my mind that "superhero" doesn't.

“Noir style” isn’t quite right though - film noir is a ’40s - ‘50s development. Pulp style is closer to the mark.

That's fair.

It's more pulp than noir.

Came here to pulp.

"Mystery Men" is a term that was used back in the early Golden Age to refer to superheroes (see, e.g. the Mystery Men Comics series published by Fox Feature Syndicate), as the term "superhero" wasn't really as widespread, given it derives from Superman.

In recent times, it has become a term used to refer to superheroes operating in the 1930s and 1940s as a reference to the usage of this term during that period.

Some authors tend to associate it more to pulp/vigilante characters, narrowing it down this style of superheroes, more similar to the proto-superheroes of the 1930s (the Shadow, the Phantom, the Clock...).