I've never heard of a code formatter called a "syntax beautifier" before. The phrase "syntax beautifier" sounds more like a compiler plug-in that actually changes the language syntax to something more beautiful.
It's a relatively common name. I would say it might even be more common than the alternative of "code formatter". I do agree that the syntax of the language isn't changing, but it does seem like the phrase that has stuck.
I think the beautifier part refers to it applying syntax highlighting to the web output.