Does Christmas (Baby Please Come Home), sung by Darlene Love, qualify? I’d say so. It mentions Christmas, “Deck the halls,” and “pretty lights on the trees.”
It was Christmas Eve, actually.
For some reason, people seem to use Christmas as the time to break out “What Are You Doing New Year’s Eve?” Logically, that song should be getting played in summer.
“Little Altar Boy” isn’t really explicitly a Christmas song, except Karen Carpenter seems to have made it one. I don’t even know if the writer intended for it to be played at Christmas.
I do know he was no altar boy himself. 
Good point. Perhaps we should emphasize that the Christmas trappings should be mentioned, but not merely in passing.
As an aside, I think that the winter songs are arguably encompassed by rule #2, since snow is traditionally associated with Christmas.
At any rate, I would now like to rephrase the criteria as follows:
[ul]
[li] It should make some sort of reference to the birth of Jesus (e.g. “Silent Night,” “God Bless Ye Merry Gentlemen,” “Joy to the World”)[/li][li] It should mention some of the usual Christmas trappings such as Santa, Christmas trees, or sleigh bells (e.g. “Jingle Bells,” “Rudolph the Red-Nosed Reindeer,” “Santa Baby,” “O Tannenbaum”) and not just in passing[/li][li] Songs about snow and/or winter that are traditionally associated with Christmas (e.g. “Baby It’s Cold Outside,” “Winter Wonderland,” “Frosty the Snowman”)[/li][li] It should mention Christmas in more than just a passing manner (e.g. “Christmastime is Here,” “The Twelve Days of Christmas”). This is especially true if it alludes to the Christmas spirit (e.g. “Do They Know It’s Christmastime?”).[/li][li] For instrumental tunes, they should be closely associated with a well-known Christmas show (e.g. “You’re a Mean One, Mister Grinch,” Vince Guaraldi’s “Linus and Lucy”)[/li][li] Also acceptable are songs that are about the spirit of Christmas and are traditionally associated with this season. “Good King Wenceslas” is included for this reason.[/li][/ul]
Or Jan 9, depending on the church.
North of the Tropic of Cancer, anyway. 
Happy Xmas is another one of those songs that despite being a Vietnam war protest song, has inexplicably become a Christmas song (apparently because it features the word “Christmas” within it).
Good thing that you amended the rules to include the “not just in passing” rule. The first verse of “Angel of Harlem” by U2 notes that:
It’s December
There’s snow on the ground
New York is lit up “like a Christmas tree”
How does the Carol of the Bells fit into the rules? It’s not originally associated with any holiday movie or show. The lyrical version (“Ring, Christmas Bells”) were not part of the original work, and the instrumental version is the one usually heard. To me it sounds like it belongs on the soundtrack to “The Omen,” which is to say, it’s one of my favorite Christmas songs.
The mark of any good Christmas song is that it was written by a Jew.
There are so many contradictions when it comes to defining “Christmas” that there aren’t a lot of rules dictating what needs to be included in order for something to qualify as Christmas-related.
Because they have a way with schmaltz?
D’oh. If I’d stopped for 5 seconds to listen to the first 2 lines in my head, I would have realized that, too:
“Met my old lover in the grocery store
The snow was falling, Christmas Eve”
That said, it doesn’t change the fact that it’s not really a holiday song.
“Deck the Halls” isn’t a Christmas song or a New Year’s song; it’s a Yule song.
What about religious songs that don’t really have anything to do with Christmas other than that’s the only time of the year a lot of folks go to church? “Jesu, Joy of Man’s Desiring”, “Ave Maria”, and Handel’s “Alleluia”, and a few others I’m not remembering right now, don’t really have anything to do with Christmas specifically.
Personally, I wouldn’t consider any of those to be Christmas songs. Except maybe “Alleluia,” but I think that one’s a stretch too.
Why “Alleluia” more than the others? That’s an Easter song.
More like an Apocalypse song, the text being taken from Revelation (19:6).
Merry Apocalypse, everybody!
There is no such thing as Handel’s “Alleluia.” He did write a chorus for his oratorio “Messiah” in which the word “Hallelujah” is prominently featured. It is not a Christmas song; it’s from the part of the Oratorio that discusses the Passion of Christ. It’s simply played a lot at Christmas because we’ve become so secular a society that we lump almost all Christian music about Jesus and his life/death into the one Christian holiday we all make a big part of our lives (Easter having long since become somewhat passe). This is also why you hear “Jesu, Joy of Man’s Desiring” (from the Cantata “Heart and Mouth and Deed and Life”), various versions of “Ave Maria”, and other such fare.
To deal with the OP, I would say that there are Christmas Carols (religious songs about the Birth of Christ), Christmas songs (songs about Christmas that aren’t particularly religious, or don’t fit the verse-refrain-verse-refrain etc. pattern of a carol), Christmassy songs (songs which have an aura of Christmas about them, though not dealing with the holiday directly) and Christmas-time songs (songs which, for whatever reason, get trotted out for the pre-Christmas weeks, but don’t have much of anything to do with Christmas or Christmas themes).
Examples:
- Silent Night
- Rudolph the Red-nosed Reindeer
- Have Yourself a Merry Little Christmas
- Winter Wonderland
Obviously, as with any such categorization, some songs will not fit neatly into the slots.
Christmas songs should have some relation to Christmas. But the bar is set really low. In fact, I’m fine with “Christmas song” being used generically like Lego and Xerox to refer to any winter season/holiday song. Or even to any song that only gets airplay in December for whatever reason.
I mean if you want to get all technical, that’s fine, for the purposes of taxonomy, in which case things can be broken down into all sorts of subgenres. But for the purposes of what gets played on radio, it’s better to have the widest possible definition so as to avoid repetitiveness.
Sure, except what actually seems to happen is that you still get repetitiveness; they’re just playing a small number of tangentially-Christmas-related songs over and over, instead of playing a small number of genuinely Christmas-related songs over and over. If we’re going to have monotony anyway, I’d much prefer the real stuff, thanks.
There’s a radio station here in Chicago, which plays “light pop” music 10 months out of the year, but which goes to an all-holiday music format from early November through Christmas. I’ve read an interview with the program director there, who notes that there are only about 35 or 40 holiday / Christmas songs which most people want to hear. They try to mix in some other stuff for variety’s sake, but if they put in too much of it, people complain.
When I first saw the title to this thread, “Last Christmas,” sprung into my mind. By the way Darren “Savage Garden” Hayes does a much better version of it.
I often hear of Dan Fogelberg’s Same Old Lang Syne played but that is a song about two people meeting but it just happens to be a Christmas, it could’ve really been any other time and still worked, never-the-less it remains one of my favourite songs by him