First off, what's a holiday? Let's see what Messrs. Merriam & Webster have to say.
1 : holy day
2 : a day on which one is exempt from work; specifically : a day marked by a general suspension of work in commemoration of an event
3 chiefly British : vacation —often used in the phrase on holiday —often used in plural
4 : a period of exemption or relief
To begin with, it's somewhat ironic that those objecting to the name "Christmas" as offensive because it "imposes religion" turn around and use a word that means (per M&W again) "a day set aside for special religious observance". Oops.
Clearly Christmas fits the first definition. Then there's the second definition (which Christmas fits into for most of us in North America and large parts of Europe, at least), which applies but has no religious connotation, per se.
It turns out that Christmas is a winter holiday; you can call it either one.
But Christmas is a specific winter holiday, meant to celebrate a specific event (Christ's birth). It has, of course, other meanings as well; the giving of gifts can mean all sorts of things, depending on what you believe and feel. The rampant consumerism, the insanity of shopping, the demand for more and more from Santa, the competitiveness of out-giving to the point of going into massive debt... these have nothing to do with Christ at all. They are simply what humans in an unfettered, western, capitalistic society have made of this day. Most of these (along with parties, decorating and the food) are fine if they are not taken to excess. They can be Christian or not, depending on who you are and why you do them. They can work for everyone.
I have something to say to both sides of this debate. The short answer is that it's (at best) very foolish to be having such arguments, especially to the point of acrimony, never mind going to court over it.
Christians: It is utterly irrelevant what anyone else calls it. The Jews don't celebrate Christmas; they celebrate Hanukkah. Are they or their faith or their relationship with God in any way lessened by the fact that you and I (and the Muslims and atheists and Buddhists and...) don't celebrate Hanukkah, that most non-Jews can't even pronounce it correctly? No! So why should we and our faith get our panties in a wad if someone else (gasp!) doesn't want to call it Christmas and celebrate the way we do? (Which celebrations, I note, are found nowhere in the Bible.)
Next, and this applies to far more than Christmas, why on earth should we expect non-Christians to get excited about Christian holidays? Unless and until they have a relationship with God, why would they care? That's like expecting a cat lover to get excited about a pit pull in their yard. (If you like both, great. Work with me.)
It's not offensive to me if someone calls it a winter holiday, Hanukkah, X-mas, or anything else. That's between them and God. I don't need to get offended for him.
Finally, dear Christians, let us remember the two great commandments. Everything we do needs to flow out of those. Love God. Love your neighbor as yourself. Our celebration of Christmas should bless everyone around us, not alienate them. We can not (and must not try to) make them enjoy it.
For those who get offended by the word, "Christmas"... you need to ask yourself, "Why?" If you get offended because of someone else's belief, that's not their problem, it's yours. And so long as they aren't trying to force you into anything with it, if it bothers you that much, then it *is* a problem, and you need to find out why and work on it.
It's Christmas. It's a Christmas tree. They are what they are. Now, frankly, I don't care what *you* call them. If it's really a statement of faith for you to not call them by that name, that's fine. It honestly doesn't bother me in the least (though I find it a bit odd in those who claim no religion). But it shouldn't bother you if I call it Christmas and wish you a merry Christmas. Take it in the spirit it's given. I will. When you wish me a happy holiday, I don't go into a funk because you didn't call it Christmas. I go, "Thanks! You, too!" and I mean it.
I'm not going to shop or not shop somewhere based on whether they have signs that say "Christmas", "X-mas" or "holiday" in the window. I'm not going to live somewhere or not live somewhere because the courthouse does or doesn't have a nativity scene.
Celebrate what you want to. Or don't celebrate. But don't demand anyone else do the same. In either direction.