That would be the "hack a KOC" approach. Spinners are known as components as rotary or optical encoders, depending on construction, which is what that bmIII turntable is using. The IIDX KOC that Konami puts out (you can find one left from the dead US release or buy one from Japan for $$) doesnt use an integrated encoder, but rather two optical gates and a plastic "toothed" disk with the gates aligned for quadrature output - identical to what a rotary or optical encoder outputs. The controller then interprets these and sends up/down button presses to the playstation.
Some people who build these things by hacking KOCs up reuse the optics and the toothed disk (basically attaching the whole thing to the underside of their fancy turntable), while others just reuse the control and construct their own encoder. I tend to build my own control because it lets me do some other useful things when hooked up to things other than playstations, lets me remap buttons if I want (gogo DJ station Pro!), and those KOC control boards (at least the US ones) have a nasty tendency to die inexplicably. Of course, making your own control is a little harder than just soldering some wires down to the right place.
You can't just hook a quadrature encoder directly up to a playstation digital controller. It doesn't work that way. A quadrature encoder generates two square waves that are out of phase with each other. You can tell how fast the device is spinning by the frequency of these square waves, and which direction by which output transitions in a certain direction (high->low or low->high) before the other one. Clearly, this doesn't look like pressing a button electrically.
There are several ways to convert such behavior into discrete events (emulating a button press), but not without a little logic. IIDX uses an 8-bit counter which is actually fed directly to the arcade software. The counter increments once per 64th of a revolution for clockwise rotation and correspondingly decrements upon counterclockwise rotation. The game then infers motion from the counter data. Newer machines may have higher resolution as 1/64th of a revolution is actually kinda terrible. That bmIII turntable would certainly be capable of much higher as the gearing makes the encoder rotate several times per turntable rotation, and most of those encoders have at least 32 pulses per revolution. The actual IIDX hardware also has a bug in the counter behavior I had to work around when I did my interfacing, but you wouldn't have to worry about that, at least.