"Christmas Music" Am I wrong?

Every year about this time hundreds of CD's get put on the shelves, and
radio stations play music - all under the heading of "Christmas Music".  It
seems to me that much of the music is really "Winter Music" or "Seasonal
Music" that has nothing to do with Christmas.  Of the remaining music, much
of it is associated with the secular aspect of the Holiday and does not have
anything to do with the religious celebration of Christmas.

I see then three categories:

Seasonal Music
Jingle Bells, Frosty the Snowman, Jingle Bell Rock, Winter Wonderland,
Sleigh Ride, etc.  No mention of Christmas and should be heard throughout
the Winter season - even after Christmas.

Christmas Holiday Music
Rudolph the Red Nosed Reindeer, White Christmas, Blue Christmas, Rocking
around the Christmas Tree, etc.  Christmas is a part of the songs, but only
the secular (non-religious) aspects of the Holiday.

Religious Christmas Music
Silent Night, O Come All Ye Faithful, Go Tell It On The Mountain, etc.  The
music that is about the religious nature of Christmas.

Is this analysis correct and are there other categories that should be
considered?

Jack