My first answer was in response to the term “unwed mom”. I don’t recall ever hearing that term used before except in literature or a movie that is a period piece.
As for the term “single mom” or even “single parent” (as mentioned in the OP), I don’t have any preconceived notion, other than that they are the guardian of at least one child and want to specify that they aren’t in a committed relationship.
The SO has two nieces and one nephew who are technically single parents if that status means that the parent isn’t married. Yet all three have been in loving relationships since long before the baby was conceived. They all live with their partner. I doubt that any of them would use the term “single parent” on the foundation of not having a legal marriage document.
Then there are other situations: a male friend who is raising his daughter without the help of his ex-wife, a friend who is raising his two (adopted) daughters alone after his wife died of cancer last year, a friend who adopted a baby when she was in her late 30s and is single, a friend who’s fiance broke it off a week before their wedding, a cousin who has five children by three different men (one out of wedlock) and is twice divorced.
All of these people may or may not describe themselves as a “single parent”. Because of them, I’ve learned not to speculate or jump to any conclusions based upon how someone chooses to describe him or herself.