Just as a suggestion; you know the standard disclaimer here, I'm not directing or telling you, just discussing this for entertainment purposes wherein I claim no skill or knowledge. You may want to replace the plug with a manual "safety switch". These are a box with a handle on the side that can be obtained for outfitting with fuses. Looks like you would something in the neighborhood of 15 amp slow blow fuses. No. 10 AWG wire should handle up to 20 amps. Get the THW or better rated insulated wire. You should install the safety switch within safe reach of the machine. This gives you a way to disconnect the lathe rapidly should an electrical problem develop. The fuses offer some extra protection.
Remember in single phase conventional home circuitry, both the red and black wire are "hot". A meter should read about 110 volts between each one and the white "neutral" wire and about 220 volts between the red and black wire. In the newer codes, there is the requirement of a separate full sized ground which runs parallel to the "neutral" (white) wire. If you want to meet any up to date and local codes you will need a qualified electrician to guide you or do the work. There is an excellent little book "Wiring Simplified" by H.P. Richter and his successors. Most big box stores have this little jewel of a book in the electrical section at a reasonable price.
Be safe, make sure the power is off, even if you THINK it is off: if you have doubts about any connections or routing, etc. stop there and don't do it. Hope this kicks things off. Geoff