The concept of a giving homosexuals marriage, but not calling it marriage was first put into practice in Denmark in '89, but called a registered partnership? So who’s idea was it to call such unions “civil unions”? Vermont appears to be the first to use that term in legislation.
It seems to be a way to remove religious context to such a union, the state is saying in effect as far as we are concerned this couple is joined, but we are not stepping on toes here and staying out of the religious issue.
A reasonable surmise, but completely useless as an answer to what the OP was asking about: namely, the specific historical circumstances of the origin of the term “civil union”.
A possible answer to the OP’s question is suggested here:
Vermont can talk all they want to about “coining” the term, but it was used multiple times in different venues from 1992, and meaning the exact same thing.
It was clearly coined by someone who had no idea how cumbersome it would prove as a verb: We’ve been unioned? We’ve civil unioned? We’ve had a civil union? (not really a good verb).
Civilly united?
Not to get argumentative in GQ, but actually “married” works! 
Civil Union is just one of many terms used to describe a legal partnership between same sex couples. It may be the default term in Vermont or the whole of the US for all I know, but it isnt universal. Here in the UK, it’s Civil Partnership.
“Married” is what I use, but you’d be surprised what umbrage it raises even among some liberals folks who claim that civil unions equal marriage.
IMHO, “civil” strips away any inplication that the realtionship is romantic and sexual.
I would disagree. I see it as stripping away any implication that it is anything but a legal arrangement. Which is really all the government should be promoting anyway.
Denmark’s Registreret partnerskab of 1989 certainly is the model for all types of these can’t-call-them-marriage contracts. I think the specific term civil union was first brought to national politics in 1991 or 1992 in France, when Le Contrat d’Union Civile was proposed in National Assembly. There union civile of course translates to English as civil union. Those became law in 1999 under the name Pacte civil de solidarité, in English civil pact. And as the French like abbreviations and that is indeed a bit cumbersome name, it’s better known as PACS, and a verb, “pacser”, was soon invented. So I would think that when reporting on the proceedings of French union civile the English equivalent civil union was widely used already in early 1990s, which certainly means that Vermont didn’t invent the term themselves.