The Christmas artifacts are not rings, they are various Christmas-related items (sprig of holly, etc.) I don't think those Christmas artifacts are available in the game anymore as I haven't seen them in a while. They had Valentine's artifacts for a while as well. But neither of those aren't needed to complete the "all the rings" objective and the ring artifacts are definitely still available as of the current (post-Valentine artifact) update.
So for completing the "all the rings" objective, it's actually good that the Christmas and Valentine artifacts aren't around anymore, as more of the artifacts you get from the chest are going to be rings. If your experience is like mine, you will get many copies of some of the rings before you finally get whatever last missing one you need to complete the objective.