As a noun, the cone-shaped opening at the upper end of a Pilot Hole, made to allow a flathead screw or bolt head to set Flush with the top surface of the piece. As a verb, using a  Countersink Punch, drive the head of a Finishing Nail or Brad below the surface of the wood in order to conceal it. James Smith, 1815, page 115: "The head of the countersink is conical."

Source: James Smith,The panorama of science and art; mbracing the sciences of aerostation, agriculture and gardening, architecture, astronomy, chemistry ... the arts of building, brewing, bleaching ... the methods of working in wood and metal ... and a miscellaneous selection of interesting and useful processes and experiments. Liverpool, Printed for Nuttall, Fisher, and Co., 1815.,

(My own curiosity will keep me working on this entry. For the woodworker, the concept of "countersink" is simple, a merely a jig designed to allow you to set the head of a screw flush with the top of a piece. "Sink" I can understand. Why "counter"? Nothing that I have examined, so far, begins to explain the reason for counter in the term, countersink.)