Good topic!
I have three distinct memories involving games at Christmas, but I don't remember as much context behind them as in your story (for my Birthday though, I have a couple stories).
Anyway, the earliest one I can remember is opening Dig Dug and Desert Falcon for the Atari 7800. I didn't have a Nintendo back then.
The second memory is playing Zelda: Link's Awakening at my aunt's place for Christmas. What an excellent Zelda game that was!
For the "final" memory, I was given Final Fantasy VI (III in the USA). I ended up playing that game for an entire year without buying another one. I had only briefly played Final Fantasy II and Final Fantasy Legend III on GameBoy, so I was completely blown away by the intricate story and the accompanying music. It is still one of my top ten games ever.