I have no experience at all with arduino or VFD. This is my first project. I read some C++ programming in school ~8 years ago. Not much and I was not great. Please keep the level low if you want to help
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The manual is not super clear, but I think I have it figured it out by now.
The figure suggests GND of the control voltage is connected to the metal parts. These should also be connected to the Protective Earth of the mains. Mains voltage can be harmful. I expect it to be safe, but I hope you'll do some measurements, starting with the connection to the PE before touching the UNO or connecting your PC to the UNO while the drive is powered.
You won't need a separate supply unit for the board if you use the 12V OUT. You can even add some circuitry to the UNO, for instance some buttons and LEDs, up to a 50mA (the board itself needs about 50mA and at 100mA the regulator will start getting warm).
For these reasons, I like to use opto-isolators for switching Start/Stop inputs. We'll use relays for industrial applications since PLC outputs are pretty tough, and it's easier to troubleshoot in the field.
May I explain why I suggested the setup I suggested.
PE (Protective Earth) was invented to provide — it's in the name — protection. But it needs to be confirmed. Both for personal safety and for compliance with local regulations.
OP (Original Poster) did not say it was to be a standalone system, but that's what I'm thinking. Using the inverter for supply of the board negates the need for an supply unit. Already having a connection for supply negates the need for optical isolation.
Using an optical isolator for the analog signal will be not very linear, or not simple. Using industrial grade units for isolation might make the connections more expensive than the items to be connected.
OP got it working using the panel so he might be familiar with parameter settings.
More info from OP will allow us to hone in on what he exactly needs.