Why is Manchester United so called when Old Trafford is in Salford not Manchester?

Like many teams, Manchester Utd have moved, so like Arsenal kept the name but were in fact based in Manchester at the time and were called Newton Heath. When they wanted a new name Manchester City was taken, and after a number of tries ended up with United, but reflects their original location rather than after moving to Salford.
Unlike Brighton and Hove, which acknowledges the division in two, you never seem to hear anyone say they come from Salford when outside the area, though it takes up most of West Manchester. I'd actually like to know why as that's where my father's from.
http://www.red11.org/mufc/history.htm

Old Trafford is not in the City of Salford it is situated just across the border in Old Trafford which is part of Metropolitan Borough of Trafford.

https://tinyurl.com/maptrafford

Salford's getting a bit more press now, but otherwise it's a bit like Smethwick being referred to as 'Birmingham'. Bootle does have quite a separate identity from Liverpool, though. Trafford Borough isn't particularly noted as a place - other than being the home of Old Trafford (football and cricket), while Salford has more history. Manchester is in the Hundred of Salford, for example. Salford and Trafford share L.S. Lowry to an extent - he was born in what is now Trafford but painted many of his works in Salford. Salford houses The Lowry gallery and centre. https://en.wikipedia.org/wiki/L._S._Lowry As to the question, Old Trafford is in Trafford, not in Salford. (I passed the MUFC railway station just three hours ago...)