There’s a lot going on there with that driver board. This photo brings up some questions about where things are connected and how.
I’m guessing the 6-pin connector at the bottom-left with red/yellow wires is the driver power supply. The connectors labeled X / Y at the bottom clearly correspond to the connections Ray asked about so are almost certainly the drive logic.
From your description it looks like you’ve narrowed down where the various motors are connected and that weird block of ceramic wirewound resistors.
With some careful testing and futzing around, I bet that particular stepper driver could be used with the Laserboard. But, if this were my build, I’d gut it and replace it with some modern digital drives.
I find it odd that X has the larger NEMA23 and Y has the NEMA17. Normally you’d see the reverse. Does the gantry beam move in the X direction (left-to-right) on this particular machine? (There weren’t any pictures of the motion control).
The Y motor can be driven directly from the laserboard’s built-in drivers with a current of 1.0A. The X motor has a current requirement of 4.2A and will require an off-board driver no matter what. Even the standalone driver Ray sells may be a bit anemic for this purpose. I’d recommend a Leadshine DM556 to go with this motor. Wire the coils in series.