I am wondering why The Equalizer was named as such.
In my opinion, the Equalizer is someone who makes things "equal", but what he does is more like a vigilante or a punisher.
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The movies are spin-offs/remakes of an old TV show. The synopsis for the original TV show probably describes it better...
The Equalizer is an American crime drama television series, originally airing on CBS from fall 1985 until late-spring 1989. It starred Edward Woodward as a retired intelligence agent with a mysterious past, who uses the skills from his former career to exact justice on behalf of innocent people who are trapped in dangerous circumstances. The series combined elements of the spy, private investigator/police procedural, and vigilante genres.
"exact justice on behalf of innocent people" is probably the salient part.