In traditional wedding ceremonies in English we used to say "to be my lawfully wedded wife". If I live with a woman for several years and introduce her as my wife she is my wife in many states in the U.S. She would be called my "common-law wife". But we were not legally wedded. No one officiated over a ceremony or signed a wedding license or had witnesses to the event. So it is possible for her to be my wife, and even my wife according to the law, but not my lawfully wedded wife. In short, I can have a wife without a wedding.
The other distinction that we encounter here is when there is a customary marriage ceremony, but the legal aspects of marriage (getting licences, celebrants, paperwork) have been omitted. The woman would then be a 'customary wife'. If someone had a wife that he married legally and another one whom he just underwent a customary ceremony with, I would say that the one he married legally was his 'legal wife', 'lawful wife' or 'lawfully wedded wife'. 'Wedded' just tells me that a ceremony was undergone, whether it was a legally recognised one or not.