What is the connotation of the word "cordial"?

Yes, I would agree. The Cambridge Dictionary defines it as ‘friendly, but formal and polite’.

In that context, I’d say “civilly and quite warmly considering their past”. To me, it’s always implied not just formality but a disinterest in making things less formal.

I go with your first definition. I see it as a formal word but meaning friendly.

amicably. Courteous is the word I always see as possibly devoid of anything but civility.

Etymologically, that “cor-” means “heart”. It implies more than a show of friendliness. It’s heart-felt. I think it has taken on an air of formality simply because the word doesn’t get used much in casual contexts.

@thorninmud got what I was grasping at.

His ex wife greeted him dressed as a bottle of medicine ( sorry ) :-/

I do not associate formality with cordiality.

Using cordially yours does not imply any future intent. I can be cordially yours today and
with passionate intensity in two weeks.

And disinterest is not synonymous with lack of interest.

I am going to see my ex-husband on T’day and will be greeting him cordially. It implies nothing other than good manners on my part and the ability to be pleasant.

Another example of the confusion in the English language. A cordial is a sweet after-dinner drink. I think @Aethelflaed has the answer with which I agree.

Cordial isn’t warm to me. It’s one of those things you’re obligated to do, forced to do and that people don’t really want to do but do because they’re ‘proper’. I can say I’m cordial with my ex-husband, nuff said.

Depends on where you live. I always subscribed to the “warm and friendly” theory. But in Australia it refers to a concentrated, artificially dyed, sticky, disgustingly sweet concentrate that when mixed with water creates a KoolAid like drink.

Remember, also, the official Entente cordiale

Friends (who speak French) with benefits.

Though I am quite inexperienced using “cordial”, with learner’s dictionary – cordial, I think, it has positive connotation.