A present that unfortunate kids like me must get when you want something that the family can only afford if you combine your birthday present money with your Christmas present money. Thus, you don't get anything for one of those holidays.
The period from about December 20th - January 5th in which a person's birthday may fall and they get a "Christmas/Birthday Present". I.E. They get one present for their birthday and Christmas instead of two due to their birthday falling so close to the holiday.
A generic greeting used for those (un)fortunate enough to have their birthday on December 25. Usually followed by: "I would've gotten you two presents, but..."
A term describing a Birthday/Christmas gift for someone who was born on Christmas Day when someone gives them one gift to serve as gift for both occasions. Usually a cheap way to get out of having to buy two separate gifts.
"I get ripped off every year! I always get a Birthmas present instead of Christmas and Birthday presents since I was born on Christmas day!"