What anchor type should I use for mounting a pull-up bar installation into concrete?
Tapcon makes a heavy duty masonry screw designed for applications like ledger boards. They come in the longer lengths as well. You'll have to drill holes with a masonry bit first (you can buy bits for a standard drill if you don't have a hammer drill), but they bite really well.

I have a pull-up bar mounted on wood studs (had to cleat it with 2x4s since the holes didn't line up). Just to be sure the cleat can't move I have 2 4" lag bolts driven in the top with 3" screws driven in the bottom. I would want a minimum of 3" on this to make sure it doesn't move.