How to wire three phases for 220V?
Well, this is a difficult area.
It is possible the engineer was mistaken and that it needs a 220V 3phase system without neutral to make 380V between phases.
It is also possible it's a 120V machine without neutral, making the phase to phase voltage quite close (but only just not) to 220V -- About 205V.
I would expect it's more likely the first one, but since my vision isn't good enough to read labels all the way from Europe, promises will not be made.
EDIT:
Looking at the motor controller you linked it's even more likely the first option, but still no guarantee can be made, since you say it's a total machine in which that controller lives, between the controller and machine input all kinds of stuff can be happening. But it is, indeed, unlikely.