OK. Glad that it got you running again. But don't forget that the Neutral coming from the outlet is always connected to the motor unless you unplug the cord. That is considered a safety No No, especially on any equipment that is plug-in instead of hardwired. If you ever have to replace the outlet or plug the lathe into another outlet, just be damn sure that it is actually the Neutral that it being connect to the motor, not just the white wire. As I wrote earlier, the switch that you have is only "legal" for 3-phase 3-wire installations.
Also, the second switch you gave the eBay link to may be a neater equivalent to the one that you have. The seller gives no information on how it is configured. Most likely, you connect the 3-phase line to one side and the 3-phase motor to the other. If that is the case and works, then it is not safe for single-phase.