Should I Say Christmas Present Or Christmas Gift?

This really depends where you come from ,since both are right, and it depends entirely on your preference.
Generally British people use present as a preference, and Americans use gift. This really doesn't matter much however, the important thing is that you give a gift, and that it is received in the spirit in which it was intended.

You can say either. Christmas present is what people usually say in the UK - the word gift is rather more common in the US - but both terms are used and understood in all the English-speaking world.

Particularly in the UK, gift can have a slightly more formal sound than present. It is more often used in a specific context such as "gift shop", "gift-wrapped" or the well-known marketing tactic of offering a so-called "free gift."

As with many English near-synonyms, the two words have different origins. Gift (which can also mean "talent", eg a gifted musician) comes from the Old Norse gipt, "give".

Present is of French origin; its use as a noun comes from a phrase "mettre une chose en present a quelqu'un" , "to put a thing in the presence of a person." So the idea of a present as something given only came into the language later; the word gift is of older origin.