They both existed, and they both married a Margaret Bate, in Runcorn. Evidently the problem is that we don't know which one married this Margaret Bate, if either, and which one married the other Margaret Bate, whoever she was.
Dixon's wife called herself Maggie but I doubt if that helps much.
Probably a case of buying the certs or hoping there'll be clues in the 1921 census.