would the half hitch on a half hitch be the same as a double half hitch?
No, Ben used that as a description of how to tie it, not as the actual name.
If you look at Ken's link it actually calls the knot at each end an overhand knot. Everyone I know starts their shoe laces with the same knot.
Basically it is an overhand knot with the other end of the rope through the middle. So each end can slip along the rope, but the knot at the other end stops it coming off the end.
This would be so much easier if you were standing here with some rope to show you.
Can you tie a Reef knot, because it is the first have of that tied at each end.