What kind of fastener do I need to attach this securely to the studs in my garage?
If you have access to the studs, rather than fastening through a wall covering, you can assure strength by sistering to the studs. 

If you use bolts instead of screws, you can make use of washers and nuts to add tensile strength where it is most needed. If you have access to the top side, you're all set. If not, rip the added segment to be narrower, you'll have room behind the piece to add washer and nut. 

To manage the full diameter of the fixture, you may have to add a sister to both sides of the stud.

Attach securely the sister piece with screws and if you want additional security, add a quality glue. I tend to the belt and suspenders over-engineering school of thought.

You're correct that the expanding anchors will not perform well in wood.