I don't know that anyone said a Carol has to be religious. Secular songs like "Deck the Halls" & "Santa Claus is coming to Town" seem like Christmas Carols to me.

Liturgically there are finer definitions separating Advent, Chrismas, & Epiphany, that can all be combined in the general seasonal celebration. I notice the Episcopal Hymnal has "Lo How a Rose" in the Christmas section but "O Come, Emmanuel" and "Hark the Glad Sound" are in Advent, "Brightest & best" and "What star us this" are Epiphany. OTOH, some of the hymns we've been doing don't seem very carolish to me, but I'm not sure what my internal compass is on this.

I was complaining about "Christ is Born" because the song has one birth verse & the rest are his life, with several verses on focusing on the crucifixion.