In West Texas as a child, I heard the greeting "Christmas Gift!" on Christmas morning rather than "Merry Christmas". Â Primarily, this was from my paternal grandmother and her daughters. Â Never from my own generation. Â Â The first person who said this to you could ostensibly claim a gift from you, if you didn't get them first. Â Â We never did actually claim the gift. It was more a humorous greeting. Â Today, one of my older cousins said her family also said "Christmas Eve Gift" on the appropriate day, to the same effect. Â Has anyone else heard of this practice?
My wife's family does Christmas Eve Gift exactly as you describe. It's not really a greeting like Merry Christmas; it's just a race to see who can claim the gift. My family never did anything like this and I am, after 42 years of marriage, still very slow to get involved with this tradition. My wife still does it with her sisters, but the next generation isn't getting into it. BTW both her family and mine are long-term Fort Worth people.
Further research - my family mentioned above was in Jones County, TX southwest of Fort Worth.  I mentioned this to my father-in-law from middle Tennessee.  He said the "old folks" used to use the same greeting at Christmas where he was from.  He is 92.