The wire needs to be 2 conductor with ground and 10 cable or better. The 40A is the breaker size for that machine when the minimum wire size is used. The manuals can be confusing so,,, if it comes with a 50 plug then it can be used on a 50 circuit
I don't own a 6 cord, use 10 on them. The breaker does NOT protect the wire in the wall for thermal, the applied or the calculated load does. However when you add multiple receptacle it becomes a general circuit the same way 120v with multiple outlets does. The only time a breaker does this is in case several items are plugged in to the same circuit.
I recently did a welder with 12/30 with 2 but each wire was a home run to the breaker, only 1 outlet to the wire. On other dedicated (except for welders) all the stuff must match. It's not for overload in the typical sense but it implies the circuit is sufficient and current limited for proper short circuit interruption
The breaker in a dedicated outlet is simply an off on switch and for short circuit, both in the circuit wire but also within the connected equipment. A buzzer has a 12 cord and legal to use 12 wire with 50A breaker. An outlet that has a 14-50 must use 6 as it is for a range or large trailer. There is no exception. You can use a 14 cord and 30a plug on some equipment. The load is limited by the machine and it's internals are sized or it has additional protection (similar to 180/200 wire feeds) to allow it on bigger circuits. Reason they list a 30 is due to the fact the minimum wire is 14. The 240 units like the 180/190 actually come 12 cord (limit for 50) to allow use on common welder circuits. The new MVP come with 14 cord but the adapter is special hence the warning about modifying it. Once it's cut off it would be limited to 30.
The breaker does NOT provide thermal to common 120v except for multiple loads. It is however current limited for short circuit. Any listed equipment is designed to tolerate or short 20 in the event of a fault. Think about light fixtures or even air conditioning and even other dedicated outlets but fixtures. 16 wire to a lamp holder. No opportunity for the housewife to come along and plug a vacuum and heaters to those 16 wires, they don't make a lamp that screws in to that base that can overheat the wire, it's thermal limited by the load but needs to be large enough to trip in the event of a fault.
AC units are like this, 40A breaker to 12 wire but they are hard wired, no outlet available for potential overload.
Power strips have a 14 cord and little reset. It's really not a breaker proper, it's a thermal overload as they have 5 recepts. It's still is only allowed on a 20 as it and connected equipment depends on the circuit for fault protection.