The problem, it seems to me, is that carol means different things to different people. Nowadays most BE speakers would take carol and Christmas carol to be the same thing. However some, very sloppily IMO, call all Christmas songs carols, eg Jingle Bells - this usage possibly has come from the USA.
For a clear description of the "correct" senses of the word, this link is excellent:
https://www.britannica.com/art/carol Here is the intro:
Carol, broadly, a song, characteristically of religious joy, associated with a given season, especially Christmas; more strictly, a late medieval English song on any subject.
My early music teacher has just gained her doctorate - mostly about the often very rude carols of the MiddleEvil period, many of which were completely unconnected with Christmas.