Looking at this again I think the starter is set up for 440v. That is what's labelled on the coil. I'll start looking for a vfd.
Ahall I'll try to explain the set up a little better. I am no electrician buy any stretch of the imagination. Also none of this was connected and running when I bought the lathe so I'm not 100% how it was connected. Incoming power goes into the starter box. Also connected to the top of the switch is the 3 button f/r/s switch, pictured below. It is the orange painted conduit coming out of the top of the box in pic 2. Separately there is a 3 position drum switch connected to the motor and old style Lima drive. I don't have a good pic but here is one of the motor/gearbox. It's hard to see but there a an orange conduit off the motor box, that is where the dorm switch is.
Yep, I see the 440V call out in you photos on the coils. Thats as good a reason as any to stop on a motor starter that is marginal.
You can use a transformer to change the supply voltage for the coils, but the heaters are likely to be an issue and finding replacements will be a chalenge. So, if you have to use it, or just want to, you can.
If you decide to junk the starter, take some time to disect it and get a full understanding of the circut. Its worth understanding these divices should you become infected with old iron disease. I also usualy salvage the heater elements from starters. They dont take up a lot of space and Once in a while they come in handy on a new aquisition, but I still use a RPC to power my shop.
Now on to the fun- lets get that motor spinning
Start at the Lima drive. How is it wired, 220 or 440? Look at the little tag above the hole the wires come in. It looks like a wireing diagram.
I will assume you need 220, so wire the motor for 220 and bring out three good leads big enough for the amperage of the motor.
Then test the motor an make sure its not toast. Use your multimeter to verify simple stuff like no connections between the windings and frame and the winding resistance is about the same between the 3 input taps. Assuming that checks out, Put the Lima drive in high gear, wrap a string around the pulley and get the motor spinning with no juce. Be sure the string pulls free of the motor. With it spinning, hit it with 220V single phase and she should go ahead and spin up. It will be noisy and shut it down in after your satisfied she is up to speed. I have never tried this test with a gear drive in place so your reasults may very but it is a very convienent way to verify a motor is good before spending time, money and effort on junk. Its wise to work on the floor or clamp the motor down on the bench when doing this. Some like to walk.
Sounds like you thinking about the VFD route to 3 phase power. If so, follow the instructions that come with the VFD and set up some nice controls that are easy to use.
With a VFD the motor starters and drum switch go to the parts bin or junk. You wont need them to start the motor or switch directions.