It looks like you have one of the newer 4-prong 220-240V single-phase
receptacles, and an older 3-prong plug. The 4-prong ones include a
neutral, so if you have an electric dryer or stove which also requires
120V for its control circuitry it's available via one hot leg and the
neutral. You should be able to buy one of the 4-prong plugs at any
hardware store. Wire the two hots and the ground terminals but leave
the neutral terminal disconnected. If your existing receptacle is wired
correctly then the W (white/neutral) terminal is the one you'll leave empty.

How many amps does the welder draw at 230V (look at the nameplate near
the cord inlet)? That plug style is usually rated for 50A, but if your
welder draws a lot less (20A or so) then that would explain the smaller
wire. See if you can read the markings on the power cord to determine
what gauge the wire is; there's always a chance that someone rewired it
but used the wrong stuff. I like to open up the cover to double-check
where the wires go before connecting any piece of used equipment like a
welder or machine tool. You'll want to make sure that the hots and
ground all go to the right places, and that the transformer taps are
properly jumpered if it's configurable for multiple input voltages.

Mike