Manchester is a funny place for this.
I’m a Mancunian, was born at St Mary’s Hospital in the city centre and have lived in Manchester and Trafford at different times in my life. I personally don’t feel I have ever lived outside Manchester at any point even though, strictly speaking, Trafford is not Manchester.
Manchester Trafford and Salford are three different boroughs. Salford its own city. And if you’d asked this question in the 1990s I’d have said that all three were different places. However like what happened in London (and Stoke, and Salford when it swallowed up towns like Swinton), Manchester has outgrown the borders of the boroughs and is a Metropolis, not a city.
In terms of the Global City Index, Manchester encompasses all areas of Greater Manchester except for the Northern towns of the county (Wigan Bolton Bury and Rochdale) as a Metropolis. Salford Trafford Manchester Oldham Tameside and Stockport is one continuous conurbation. Anyone calling any of those areas “Manchester” wouldn’t be wrong apart from in a sense of the outdated made up 1974 borough lines on a map.
So Old Trafford isn’t in Manchester but it is.
As for Bury, due to the geographical nature of the conurbation, it is not part of the Manchester conurbation because of the definite disconnect of Wigan Bolton Bury and Rochdale from that conurbation. However, Bury is part of Greater Manchester, the Greater Manchester Built-up area and the Manchester Statutory City Region (which actually also includes Warrington and Macclesfield as well as other areas). So Manchester Lions? Fine by me!
Does that make sense?