I've put VFD's on two of my lathes and really like having the control they provide. The other type of phase converter is convenient for when you want to keep the controls stock and don't want to get into a big conversion right away.
The main thing to know is VFD's have to be hooked directly to the motor. That means any controls on the machine itself won't work until you integrate them with the VFD. If you like you can controll on/off, fwd/rev, and speed directly from the VFD box but if you want to use the levers and buttons on the machine itself you'll need to read up on control systems and modify what came with your lathe.
If you want to see what's possible you can look for threads here by/with
@mksj, lots of us have converted machines but he's the resident expert and always seems willing to help.
But, basically you can disconnect the motor from any existing controls and just wire the three power lines to your VFD to get the machine up and running. Tons of threads on here about conversions both simple and complex but honestly if I was working on that machine I'd go with a static converter first. Reason being is there appears to be lots of controls, and the static converter will be the quickest way to power it up and asses it's condition. Good thing with a static is you can run with it for a while, then change it over to a rotary converter later on with the addition of another 3 phase motor if you choose to go that way.
So, I'd definitely recommend doing some research on the differences between VFD's, static and rotary phase converters. I've done business with North American several times and have been happy with their products and service. It looks like your machine may need some evaluation before you decide what direction to go with it. The static converter is the quickest path to three phase from your household 240v single phase, you can spend your time working on the machine itself rather than trying to figure out control systems, provided you eventually want it working as designed.
If you've already bought the VFD and want to see how the gearbox sounds after you do an oil change and cleanup, all you need to do is disconnect the motor and wire it straight in. But, unless you're ready to dive into some serious electronics before even working with the machine the static would be the way to go IMHO.
John