It's amazing what you learn on the Dis!!! This is the first place that I had EVER heard of "Santa gifts" vs "Mom & Dad gifts". Growing up, whatever was under the tree and in your stocking was from Santa. And everything under that tree was wrapped.
The ONLY time I ever remember something NOT being wrapped was one year my parents got us each a punching bag.
(you know, the kind that you blow up and you just punch away, not the kind that pro boxers use to warm up LOL) My parents blew them all up ahead of time and had them standing next to the tree. Boy, let me tell you, when my brother, sister and I got up in the middle of the night to scope out what was under the tree and we rounded the corner into the living room and saw the shadows of 3 small figures next to the tree, we about laid a golden brick, right then and there. 
We thought they were some of Santa's elves. LOL
Anyways.....so when I had kids (they're 13 and 15 now) Santa brought everything, everything was wrapped and I think maybe a couple years everything in the stockings were wrapped. So some years I did wrap everything, some years I didn't. This year, I didn't.
I know it's been brought up on here before about getting movies or video games or things that like that to put in the stocking. Some people think it cost too much $$$ to put in a stocking but the way I think of it is.....if it fits in the stocking, it'll go in the stocking no matter what the cost. And those gifts were ALWAYS wrapped before going into the stocking.
As for having special Santa paper, I've never heard of that before either. I just buy whatever paper strikes my fancy and wrap away. And EVERYTHING goes under the tree once everyone is in bed (my youngest actually told me to keep up this tradition. He said if he sees ahead of time what's under the tree, it just spoils the surprise) My oldest of course, could give a hoot....he's been begging me to open his stuff all week. But to each his own, no way is the wrong way, whatever works for your family is the right way. So just keep doing what you do and don't worry what anyone else thinks!!!
Merry Christmas!!!