Countersinks are available in a range of angles - the angle being the included angle of the countersunk hole. I have 60, 82, 90, 100, 110 and, as I recall, 120 degree countersinks. If the included angle of the screw head in the Chambers kit is 60 degrees and you countersink the hole with an 82 or 90 degree countersink, the screw head will make contact at the bottom of the countersunk hole, but will leave a bit of a gap at the top. Conversely, if the screw head has an 82 or 90 degree include angle and you countersink the hole with a 60 degree countersink, the screw head will only make contact at the top. Better looking situation, but still not good. I would say get a 60 degree countersink if that is what Chambers calls for. When it comes to countersinks for installing the touch hole liner, the contact area is so narrow that if an 82 degree countersink was all you had, and it should have been 60 degree, you should be able to lightly peen the liner to fill the gap. But why would you do that if you have to buy the countersink anyway?
Perhaps Jim will chime in.