Christmas carols like “Hark! The Herald Angels Sing”, “Angels We Have Heard On High”, “O Come, All Ye Faithful”, “Joy to the World”, etc. are considered Christmas hymns, because, well, they proclaim the birth of Jesus Christ and they are considered hymns for that reason. Most religious Christmas carols out there are only the said above (of course more) but what else would we use?
Yes, except the ones like “Jingle Bells”, “Rudolph the Red Nosed Reindeer”, and “I Saw Mommy Kissing Santa Claus”, since they don’t refer to the birth of Jesus.
In America, any Christmas song has become a “Christmas Carol”.
The original carols were dance songs and not limited to Christmas, as can be seen in the great collection Oxford Book of Carols. The very word “carol” comes from the Greek “choros aulos”–to dance with flutes.
“Hark, the Herald Angels Sing” is a Christmas HYMN.
“Frosty, the Snowman” is a secular SONG.
“I Saw Three Ships Come Sailing In” is a Christmas CAROL.
And that hackneyed, horrible “Lord of the Dance” is, oddly enough, a carol, as is the original “Simple Gifts.”