Why do so many people say that the Father gave us Jesus as a Christmas present?

Wasn't the giving of the Son done at the cross, not at the manger?

John 3:16 For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.

Wouldn't the following show the manger with: "unto us a child is born", but the cross with: "unto us a son is given"?

Isaiah 9:6 For unto us a child is born, unto us a son is given: and the government shall be upon his shoulder: and his name shall be called Wonderful, Counsellor, The mighty God, The everlasting Father, The Prince of Peace.

If the giving of the Son was at the manger, and was a Christmas present to the world, then why did only the shepherds and wise men know about it?