Mrrzy: The only problem with calling the celebration "Midwinter" is that the last days of December occur at the beginning of the winter season, when the days actually begin to lengthen after the solstice and the weather isn't yet at its coldest. Since, as you say, "the point of [your] family midwinter celebration is love-demonstrated-by-giving", I'd suggest calling it "Giving Day."

I see no problem with having a tree, holly, traditional foods, even figures of people with wings (hey, fairies have wings, too!) without attaching any symbolism to them whatsoever. Sometimes, a tree is just a tree and a present is just a present. (...and a cat... is a person! Sorry.) Whatever trappings and traditions bring you comfort, leave 'em in!

I have to admit, though, that I'm disturbed by the concept of "limiting" the singing. Certainly, if people feel uncomfortable singing a song because of its lyrics, they should not. But the "message" of a song is a more subjective concept; some of the narrative Christmas carols could be considered ballads or otherwise folklore-oriented ("The First Noel", for instance), whereas those that are songs of praise (such as "Joy to the World") would not. And any song that expresses a wish for peace and goodwill can't be all bad, in my book (speaking as an agnostic).

I would say, sing whatever feels comfortable to you since, in the context of your family celebration, no one is going to "take it the wrong way" (take it to mean that you believe something that, in fact, you do not). And if the song everyone's comfy with is "Sweet Betsy from Pike", who cares? The important thing is that you're gathering together and singing!