I don't know if this will help. Here's the inside of my lathe control box. I have 220VAC single phase in. Output from VFD is 3 phase to the 1HP lathe motor. Schematic attached too.
I'll prolly have the buss bars, and the VFD but no relays or transformers.
I don't have a coolant pump or an ELS nor do I want them. Under normal circumstances, I'd love to run my lamp the way your is done, but I have learned to totally completely HATE the factory 24V lamp. It burns out if I burp.
So I want to replace it with a 120 or better yet 240V AC LED lamp. Maybe two 120s in series so they can be run off of the 240 input. That is project 42k.
I found a Grizzly schematic for a 240 3phase lathe that I used to get going a month or so ago, but quickly realized I was better off starting from scratch.
I am currently busy roughing out my constraints and operational requirements. Then I'll do a rough schematic, and then start sourcing parts - which will dictate some changes. The wiring itself should be relatively easy after that.
Thanks ever so much for thinking about me and reaching out to help.