Clare
I too agree that there is not an error in the records - the Registrar recorded how the birth was registered. So there is nothing to be changed. And, as others have explained, you do not have the wrong surname.
I have several people in my tree who knowingly or unknowingly gave the "wrong" surnames in B, M & D certificates. It is all part of the fun for me in unravelling who did what and to whom...
I call myself a family historian, rather than a genealogist, as I am interested in the family I came from in the past 200 years, and what they did, rather than what my bloodline is. But there again, my lot all have names which are common, and belonged to the lower orders, so I doubt if I could follow my bloodline very far.
But other people care a good deal about their genetic origins, and I guess they are genealogists.