You would first have to decide whether you want "cheap", or "quality and reliability".
My preference would verly likely be for the mains voltage powered big box. I assume it's got a mains voltage powered transformer in it and a triac to control it.
The other smaller box has a small wallwart in it, which means it is probably built around a few Li-ion cells and MOSFets, and pulling such high current peaks from Li-Ion is far beyond the recommended operating conditions of those things, but lot's of people do it and apparently it does work. A lot of welders of this type do have a serious flaw. During the high welding current, the battery voltage sags, and sometimes also the gate voltage of the FET's. When this happens, the FET's can get partially closed, which can increase the dissipation in the FET to instant destruction. If you have a welder of this type, then always do a check of the Ugs voltage during welding. Adding a small diode and capactor can buffer the voltage for feeding the electronics during the welding pulse. But I assume that the better versions have a step-up regulator to make an Ugs of 12 to 15V. (This usually halves Rdson compared to an Ugs when the FET's are just opened).