I think you are talking about 3 wire control with regard to momentary switches, the default is 2 wire controlled which uses maintained switches (page 4-34). The issue with 2 wire controlled using maintained switches, is that if there is a power outage or you accidentally leave lever in the forward/reverse position, the machine will restart which presents a safety issue. Using the VFD, you usually would gut the old control board, since it would not be used giving you space for other options. On a lathe, there is usually a power relay that latches either with a button or when the spindle direction switch is in the stop position, this prevents a restart. An E-Stop would normally interrupt the connection to the COM input terminal (or the power relay), the danger with maintained switches, is that the late would restart when the E-Stop is reset if a power relay is not used. Killing the power to the VFD with the E-Stop or via a contactor is not recommended in this setting, and a simple power reset relay is more optimal.
The Teco does have a feature which prevents auto restart, on page 4-58 : 07 - 00 would be set to 【0】: Momentary Power Loss and Restart disable. You could try this with 2 wire control and see if that works. One of the inputs can be reprogrammed to JOG if needed. I have some schematics for relay based controls, but it is a bit more complicated.
What is the specific Jet model you have and what functions do you want?