Hi Joe,
So all of this stuff about wire gauge is a bit confusing, but let me try to set your mind to rest a bit. As, I said before, I have a little different machine than yours, but the basics are the same. The cabinet for my CNC version does not look anything like yours so I will not post it here, but there are other pictures already posted by others. What I am posting for you are a couple of pictures of my VFD wire wiring and the cabinet power switch. By the way, my motor is 1.5Kw which is more like a 2HP not 3HP.
I think you are trying to follow codes on wires and that is good. (But not all wires of the same gauge can carry the same current. (But a bit more about this on down.) Also, the current per wire of the three current carrying wires for three phase can be different than the current per wire of the two current carrying wires, in single phase. So think of it this way , the "load" on a 220 single phase circuit is essentially the same as the load on the three phase circuit, i.e. it is the same motor. There is some inefficiency in the VFD so it takes a little power also, but not a lot. They are pretty good at doing the conversion. So lets assume the load, Watts, are the same, power as it would be with or without a VFD. Power is defined as P=IV= single phase power = three phase power, where the single phase power is being carried to the load with 2 wires and the three phase power is being carried by 3 wires. So the current value in the three wires is a smaller value per wire, than current carried in each of the two wires of the single phase wires. So naturally the gauge of the wires could be different. The 3phase wires coming out of the VFD to the motor could be smaller than the single phase wires going into the VFD.
But there is yet another twist. Not all wires of the same gauge are equal in their current carrying capacity, because not all wire insulation are equal. If you look at a the wire insulation you can usually find some letters on it, indicating its ratings. Normally we wire our house with romex NM wire. The NM indicates the code you are thinking of. i.e. Single phase, two wires, 220V 30A wire is 10 gauge. (20A is 12 gauge, 15 A is 14 gauge, etc.) There are lots of other insulation ratings for higher voltage( electrical break down) and for higher operating temperatures (decomposition or melting). Meaning that 30A can go out one wire to the load and return in the other wire. It does so over reasonable distances without a substantial voltage drop along the wire. The voltage drop along the wire tells us that some Watts are being dissipated along the resistance of the wire heating it up. (Ohm's law V=IR or for power, P=IV=I^2R= V^2/R) . Since the wire has length this is dissipation per length. The longer the wire the more power is lost. So the logic is that there should not be too much loss in the wires or there will be no power left for the load! Likewise, the heat per unit length should not get the insulation so hot that it would fail.
Inside the cabinet, the wire size and insulation are most likely different from that which is bringing the power to the cabinet. It also matters that the wires in Romex are next to each other and encased in the outer insulation where as the wires inside the cabinet are separate most of the time. Meaning that their heat can escape to the air easier. Anyway, It is common that inside the equipment one changes the wire gauge and insulation ratings so that smaller gauge (diameter) wire can be used to bend and to make connections. For, example, if you open up an electric oven you will find that the wires going to the oven maybe 220V 2phase at 20Amp, but once inside the connection box the gauge maybe 16 or even smaller diameter, but will have a higher temperature rating. This wire is inside the oven or metal conduit so its elevated temperature does not touch much of anything.
By the way , it was mentioned the current draw can be different when the motor is starting, and definitely is higher if the motor is stalled. This is because the impedance (dynamic resistance) of motor changes. (related to the term "back EMF"). Circuit breakers are designed for a certain amount of this as they allow short transient time periods of extra current before they get hot enough to trip (think slow blow fuse). Likewise, wires can over heat for short periods before it becomes a problem.
Lastly, there is never anything wrong with having too large a wire carrying capacity .... if it makes you feel better... it just cost more $$. You just never want to put in a breaker that is rated larger than the current carrying capacity of the wires that you use ... as described above, capacity is not determined by gauge alone!
Dave