In my reading, Vic's post refers to making centre holes for the purpose of driving using lathe dogs, predating the use of scroll chucks. Martin Kyte mentions four jaw chucks in passing, and also the drill press.
Going down the rabbit hole (and attempting to answer my own comment about Slocombe drills), US patent US2481939A (applied for 1945, awarded 1949) shows a modern style centre drill. US Patent 505,013 (1895) mentions a spade-type drill (and also a twist drill or straight fluted drill) to form the centre.
So I might presume that up to that time, some craftsmen might make their own flat centre drill. In the 70's I made a flat drill from a forged end on 1/4" silver steel, turned and filed the end, then hardened and tempered. About 3/8" diameter, I was surprised how well it cut into solid steel without a pilot hole. Most of us are happy to grind our HSS to suitable shapes, such a centre drill is little more work. And if we don't have 'sharp' centres in our lathes, we wouldn't need the little hole for clearance of the point. OK, a pocket for oil, grease or (remember this) tallow.
Bill