The whole 10mm=41Mag nonsense started back during the early days of the 10mm.
During that era, one of the more common 10mm loads from a major ammo maker was the Winchester 175gr STHP, and a popular (if that can really be the right word to describe any .41Mag ammo) .41Mag self-defense loading was, coincidentally, also a 175gr STHP from Winchester. The two loadings provided roughly similar velocities, in fact, the 10mm loading velocity was advertised at 1290fps (it's currently catalogued at 1200fps) and the .41Mag was advertised at 1250fps.
But 1250fps isn't anywhere near qualifying as a hot .41Mag load--in fact it's quite a light loading. One of Keith's design concepts for the .41Mag was that it was to be loaded in two general categories. One category was an easy shooting self-defense cartridge, the second second general performance category was to be a hunting round, or at least a very high performance heavy magnum round that handily exceeded the common service pistol calibers of the day. Remington took that idea but heated up both of the categories a bit from Keith's vision.
Anyway, the .41Mag 175gr STHP from Winchester clearly fell into the "mild" .41Mag category intended for self-defense/LE use which, though hotter than what Keith envisioned, was still far below what the caliber was capable of.
On the other hand, 1290fps with a 175gr bullet was pretty hot for a 10mm round. Not the absolute top end, but certainly not anywhere near the bottom.
But some gun writer did the comparison and without getting into the details of how the two loads compared to other loadings in the two calibers, gushed about how the 10mm beat the .41Mag. I remember reading the article although I can't remember the author now. It's been "common knowledge" ever since.
In reality, if you go to the trouble to catalog a ton of loadings from the 10mm, .357Mag and 41Mag and then compare them, it becomes apparent that the 10mm and .357Mag are (overall) close enough that it's hard to pick a clear winner and even more apparent that .41Mag beats the pants of both of them. Its in a completely different class.
Here are some charts I made from all the factory .40, .357SIG, .357Mag, 10mm, and 41Mag loadings I could find on the web back in 2006.
https://thefiringline.com/forums/showpost.php?p=2672982&postcount=11
At that point in time, the highest momentum factory 10mm round beat the highest momentum .357Mag loadings slightly while the highest energy .357Mag factory round beat the highest energy 10mm loadings slightly. The two rounds are much more similar than they are different.
It's interesting to see how the 5 calibers overlap in some ways and are clearly distinct in other ways. It makes some things very clear. .357Mag and .357SIG are quite different in capability and performance. .40S&W and 10mm are quite different in capability and performance. 10mm and .357Mag are very similar in capability and performance with the primary difference being that it's possible to find much lighter .357Mag loads--most likely because anything "cycles" in a revolver while the same is not true of light-loaded autopistol rounds.
Of course, it's a pain to gather all that information, poke it into a spreadsheet and then generate plots, so most people content themselves with comparing a few cherry-picked loadings or maybe just relying on opinion and hand-waving alone.