Looks Right
It looks to me like you have it layed-out correctly. Your drawing showing the hook-ups is a little small and I little hard to read, but looks correct.
What you have here is a Delta to Wye connected transformer. The primary side is connected in a delta configuration. The secondary is connected in a Wye configuration. The point where all the wye legs come together is the X0 point on the transformer diagram.
The only item that I am not certain of is the grounding of the X0 point on the secondary of the transformer.
The way that you can get 277V power out of the secondary (as it is noted on the transformer data plate) is to use that X0 point. If the primary is hooked up to 600V power, you would see 277V from that X0 point to any of the X1, X2, or X3 points. You would also see 480V power between any of: X1 to X2, or X2 to X3, or X3 to X1. These are the points that you want to hook up to the machine - as you have drawn.
There are applications where you want that X0 point to "float" and not be grounded. There are applications where you don't want that point to float and you want it grounded. I believe that you can leave it gounded as it currently is, but I need to research that a little more. Maybe some else on the forum can comment on this.
I would hook it up as you have shown all the way up to the output of the transformer and then try it. I would measure the individual voltages between X1-X2, X2-X3, and X3-X1 by having the power disconnected and hooking up a volt meter - THEN hooking up the power and trying it. Repeat for each leg. DO NOT TRY TO MEASURE THE VOLTAGES WHILE HOLDING THE TEST LEADS AND MOVING THEM AROUND!. (see next paragraph) If you are getting 480V as mentioned, then hook up the transformer to the machine switch and try it. Use appropriately sized wire. Calcualte the amps and pick the correct wire size from a wire size chart.
As a previous poster has mentioned - a "zap" from 110V is annoying. A "dry" zap from 220V will hurt (stupidly, I speak from experience here). A "wet" zap from even 220V can be fatal (thankfully I haven't experienced this yet). Almost any zap from 440V and higher can kill you (lots of stories). It only takes 1.0mA of current through the heart to kill you. That is 1/1000 of ONE amp. It's not a lot of current. Assuming the average body resistance is somewhat constant, any increase in voltage makes the chance much greater that a lethal dose of current will pass through the heart. So, disconnect the power ANYTIME you are playing with wires in high voltage systems (OK - ANY system). Or, have the electrician do the "dirty work."
Let us know when you get it set up and your machine is running.
R/Todd