This term doesn't seem to be used anywhere other than here and other boorus, which copy our "hard translated" tag. Our own wiki uses both "hard translated" and "hard-translated".
The terminology is derived from subtitling, where there exists a distinction between soft subtitles (softsubs, which are rendered separately from the video for which it is intended) and hard subtitles (hardsubs, which have the subs directly in the video), spreading through fan subbing spaces. With the origins of the site in mind, making the leap from 'hard subtitles' to 'hard translation' as a neutral term once notes became a thing wasn't too much of a stretch (further proven by the 'hardsub' alias the tag has).
To my knowledge, there isn't any neutral term which exists that encapsulates what the tag is supposed to entail, because most platforms do not have 'soft subtitles' for images in the same way the note system works for booru websites - translation means having the text on the image, not separately. The BUR here intends to just improve the grammar of the tag per English rules.