Sberry, I've felt your frustration as I read through all of this. I have tilted at the same windmill a couple times in the past - maybe on here and also probably on another site when I was trying to learn about this too. You initially are told by the experts that it's against code, then when you show them the code that they didn't know about, they have a different reason why it is wrong. Small welders have duty cycles that won't allow them to draw enough prolonged current to overheat that smaller sized circuit to the point of melting and causing a fire. Common sense should tell anyone that when the manufacturer puts a 50A plug on a #10 or #12 cord, that you don't draw a constant 50A. regardless of the fact that power cord wire & insulation will allow a smaller conductor size compared to the branch circuit in the wall, it's not the reason they can use a smaller conductor. if so, that type of insulation would be used everywhere to reduce conductor sizes, because plastic is a lot cheaper than copper.
I'm sure that those in here who are professionals in the field do a great job and are very careful about what they do, but the last time I hired a well known and respected local professional was to wire in a transfer switch. Flash forward a few years later when I got in the box to replace a weak breaker I immediately saw one connection out of the 10 circuits for the switch was just twisted and black taped with no wire nut. been that way for years and probably would have been fine forever, but that doesn't make it right. I even got home just as he was leaving. A true professional could have told me he ran out of wire nuts and had to go get more... heck, i had plenty there that could have been used to remedy the issue in 5 minutes, but they chose to bury the issue and leave. I read and try to learn how to do things correctly partly because it's just my nature, and partly because there's very few trade professionals that I trust explicitly. when what I read conflicts with what the professionals tell me it makes me a little edgy...