Doug Fattic 

Yes the counterbores in your picture is exactly the bit you need. If I were you, I'd secure the little bench press to the bench top somehow so it can't move. Probably with a C or any other kind of clamp so the drill press stays put. I'd also make a special effort to secure the fork to the drill press too. You can position the fork before clamping it down somehow by putting the pilot of the counterbore in the enlarged brake hole first. When everything is very secure you can drill the counterbore.

Does your drill press have a stop on the downfeed? You need to make sure you don't drill too far. That look is terrible. Somehow you have to make sure to drill it far enough but not too far. If you don't have a stop, you can put tape on the counterbore so show you how far to drill. Remember that before you remove the clamped fork, to drill the piloted hole out to 8 mm (which is almost exactly 5/16").