What wire gauge should I use with a 30 amp breaker for a 220V welder?
The calculator you referenced is only appropriate for voltage drop and makes the disclaimer that the results of the calculation do not account for current specs or NEC. Any circuit fused for 30 amps must use a minimum of 10 ga copper or 8 ga alu. Longer runs may require an upgrade of wire size.  In your case, use at least 10 copper for your welder regardless how far it is from the breaker panel. 
I'm sure one of my buddies here will have a NEC ref for ya.