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.
Wiring from garage panel to welder
NEC 630.11(A) may be of interest.
National Electric Code 2008
ARTICLE 630 Electric Welders
II. Arc Welders
630.11 Ampacity of Supply Conductors.
(A) Individual Welders. The ampacity of the supply conductors shall be not less than the I1eff value on the rating plate.
Alternatively, if the I1eff is not given, the ampacity of the supply
conductors shall not be less than the current value determined by
multiplying the rated primary current in amperes given on the welder
rating plate by the factor shown in Table 630.11(A) based on the duty
cycle of the welder.
Also take a look at 630.12 Overcurrent Protection.
The Owners manual (PDF) seems to contain the answer.
Wiring from service panel to sub panel
If we look at 110.14(C) in 2008 NEC, it says:
National Electrical Code 2008
ARTICLE 110 Requirements for Electrical Installations
I. General
110.14 Electrical Connections.
(C) Temperature Limitations. The temperature rating associated with the ampacity of a conductor shall be selected and coordinated so
as not to exceed the lowest temperature rating of any connected
termination, conductor, or device. Conductors with temperature ratings
higher than specified for terminations shall be permitted to be used
for ampacity adjustment, correction, or both.
Which means we'll have to check the breakers temperature rating, and the subpanel terminals temperature rating. Then we'll have to look at table 310.16 (Table 310.15(B)(16) NEC 2011), so we can determine the wire size we should use.
In this example image you can see if we're using copper, we'll use 60°C. Once we look at the table, we'll see that for 50A @ 60°C we need 6 AWG copper wire. (50A @ 75°C = 8 AWG).
110.14(C)(1)(a) basically says that if the temperature is not listed, use 60°C for circuits less than 100A. 110.14(C)(1)(b) says if the temperature is not listed, use 75°C for circuits over 100A. However, in the real world 75°C is often used for both.
Best Answer
Yes, the wire is undersized for 20 amps - 12 gauge copper is code minimum for 20 amps - if the run is long, 10 gauge can be a smart choice, if it's very long a larger size feed (30 - 60 amps at 240V) to a subpanel run on aluminum wire of much larger size quickly makes financial sense given the relative prices of aluminum and copper.
The cycle in effect is that the motor draws a large amount of current when it starts, and that drops the voltage on the undersized wire, and that makes the motor want more current to maintain power. Since heating goes up as the square of current, the motor overheats and shuts down. Keeping the voltage up with adequately sized feed wires keeps the current more reasonable.
Rewiring to a 240V subpanel in the garage may also offer the option of switching the saw to 240V operation, which will reduce the current by half (and the heating by 3/4) so it will start more easily. Depends on the saw motor, but most have that option.