A #8 countersink will not only drill too small a hole, it'll also leave a #10 screw's head protruding slightly above the surface (assuming flat-head screws, since countersinks aren't used for any other screw head style). Use a #10 countersink for a #10 screw.
From your description of the screws you intend to use, though... I don't think you want to countersink at all, only predrill. If you need those heads to be below the surface, you'll need to "counterbore" instead of countersink; counterboring leaves a flat-bottomed hole instead of a taper-bottomed hole, and it's done with the appropriate size Forstner drill bit... BEFORE drilling the clearance hole.
You may or may not be satisfied with the tapered drill included with that countersink for drilling both the through hole and the blind hole (the one the screw bites into). A #10 wood screw starts and seats very well into a 9/64 blind hole made with an ordinary straight-shanked drill bit.
Since you're driving into hardwood, you'll do yourself a big favor if you go pick up a "toilet wax ring" from the hardware store and use that wax to lubricate your screws before you drive them. The difference in driving effort is amazing, and the screws are far less likely to split the wood. Don't use soap as a screw lubricant - it contains water and will rust the screws (unless they're brass).