Does one of the designations indicate that one of the partners is dead vs. divorced? First vs. Former
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"First wife" is mostly used if there is a second (or further) wife. An exception would be if there is a question about wifes, like "Has he been married before?" "No, Jane is his first wife". There is no indication whether that wife is alive or dead.
"Former wife" is used for a person who was once the wife and isn't now. It is usually not used for a person who died while being the wife. So the "former wife" would be a divorced wife, with no indication whether that former wife is alive or dead.
So if Joe marries Jane, she dies, and he marries Joanne, then you would call Jane his first wife, but probably not a former wife. If Joe and Jane divorced then Jane would be both the "former" and "first" wife. Before Joe married Joanne, Jane would likely not have been called his "first" wife. Joanne may at some point become a "former" wife, but she can never be the "first" wife.
If you met Joe and Jane together, and Joe said "this is my first wife, Jane" then you could reasonably assume that they are divorced and Joe married again. If Joe said "this is my former wife, Jane" then you could reasonably assume that they are divorced, but you couldn't assume anything about Joe being married again. And of course people use language in an imprecise way all the time.
First only means that the speaker/subject has been married more than once. The spouse in question can be deceased or alive.
Former is the more formal equivalent of ex-; both indicate divorce.
There is no accepted term for a person to whom one was once married if the marriage ended in annulment, because the marriage never existed.
Late indicates deceased.
This is my late wife's first ex-husband.
This means that the speaker's deceased wife married more than twice, divorcing at least two.