What you are referring to here is called the ordeal by hot water (by modern classifications), to be differentiated from other ordeals such as the ordeal by fire, trials by combat, ordeals by hot coals/irons, ordeals by water, ordeals by bread or cheese, and more! This ordeal can then be further differentiated into the ordeal by the first degree or to the third degree. The difference between the two was the degree to which the person had to submerge their arm into the boiling pot. The first degree stipulated submission to the wrist, while the third degree involved submerging your arm up to the shoulder to retrieve the stone or weight. However there is a great deal more to unpack in the history of the ordeal and ts use in determining guilt (or innocence!) to let's unpack some of this below!

The ordeal is a bit of an unusual legal entity, as it fluctuated a great deal in terms of acceptance, popularity, and utility. It has its origins in the legal systems of the Germanic peoples that became the new ruling powers in much of the former Western Roman Empire. While not totally unique, the prevalence of torture in the Roman legal system has been compared to the systems of ordeal in the emphasis on physical pain in investigation/punishment that characterized Late Antique/Early Medieval justice, however it did not come into its own until the period of consolidation of states in the former Western Roman lands. The realms of the Franks and Anglo-Saxons in particular, later on France, England, and Germany, became a kind of cradle for trials by ordeal. In Italy, Iberia, and southern Gaul Roman legal institutions were more robust in their survival, to a certain extent.

However there are a few things to keep in mind. In general, most legal suits likely never made it to the stage of needing an ordeal to determine wrongdoing. This is because of the nature of judicial investigations in the early Middle Ages. In short, prestige and social standing determined your legal culpability. If you were accused of a crime, say slander, but could summon enough character witnesses to attest to your moral uprightness, then that would usually be the end of the matter (unless you could be proven to be lying which caused a whole host of other issues), this went for other crimes as well. The only way that a suit would move towards an ordeal would be when there were not enough witnesses to testify to your character, a man refused to swear an oath in his own defense (or was not allowed to for past transgressions) or perhaps if there was other compelling reason to do so (the clergy for example in England were often excepted from being able to call witnesses or pay fines [or be paid fines] so they had alternative means of being punished/supported by their community).

Being considered untrustworthy, not being able to swear an oath to your innocence, or not having enough witnesses to testify on your behalf, or any combination, would force you into take the more punishing ordeal, such as the third degree ordeal. Note that this was not the punishment. This was a means for testing guilt, and if you failed an ordeal the punishment would then be in effect. This could be as simple as a fine (or wergild) or it could be more demanding, if you failed an ordeal when accused of treason, murder, repeated thefts, or something else severe, the penalty could be physical mutilation or even death.

Even early in the Middle Ages not all forms of the ordeal were well accepted, and even by the 9th century some forms were falling out of favor. As the Middle Ages wore on, by around the 11th century, there were a few common forms that it took.

There were the ordeals such as the corsned, or literally the bread or cheese ordeal, which at least in England in the 11th century was an option for members of the clergy who had been accused of crimes, it involved the consumption of blessed bread and cheese. if the accused ate it without trouble, they were innocent, any issue in swallowing was indicative that the accused was guilty. This form of the ordeal, at least from the sources that I have seen, was more for members of the clergy who were accused.

There were others as well such as the ordeal of fire/hot irons/coals. In these cases the accused would undertake some form of penitential walk over or through burning coals, or between flaming pits. This is one of the more well attested forms of the ordeal, and one of the last to be rooted out. Emma of Normandy in the 11th century, Peter Bartholomew in the 12th, and Savonarola in the 15th were all subjected to it. The lack of injuries on Emma of Normandy was taken as evidence of her innocence, while Peter and Savonarola weren't so lucky. (Peter died of his injuries walking between two walls of fire, and Savonarola chickened out of his trial and was subsequently arrested).

Other ordeals included dunking, trials by combat (not done with until the 14th century in France, the 16th in Ireland and Italy), and other methods of determining truth.

The ordeal of the first/third degree I've already mentioned, but I want to deal with the part of your question that looks at the reports of injuries, or lack thereof, on those subjected to ordeals. There are also stories, dating back to the period of Roman persecutions, that highlighted the ability of saints, particularly holy people, the innocent, and martyrs to be spared from the heat of red hot irons, boiling water, or other burdens who were able to bear their burdens with no injuries. These were later common too among Medieval hagiographical writings that detailed the lives of various saints. Indeed we should really see the idea of people being able to carry red hot irons without injury as a literary trope that was repeated and recycled in many different contexts. The actual veracity of these stories, at such a distance of time and given our source materials, is really a matter of faith.

Now modern science and logic generally doesn't give much credence to the idea that divine intervention was responsible for the lack of injuries on these people and seeks answers. Sadly this is a largely fruitless endeavor since there is nothing to really base alternative ideas on. I'm sure that you can google online and find ideas that some people have proposed for being able to bear the heat of stones and irons without injury, but these are simply modern explanations in the lack of any compelling, to modern sensibilities, evidence.

Over time however the ordeals were restricted, largely at the urging of the Church. Judicial duels, ordeals by fire, water, and stones, and like were restricted as the Middle Ages wore on in many cases, only resurrected later during the Reformation and Counter-Reformation, particularly in witch hunts, but it is rather clear that during the Middle Ages themselves, the Church restricted and eventually banned many forms of ordeals.

This has led some scholars, such as Peter Brown, to reconsider the role that the ordeal actually played. He indeed argues that the goal of most ordeals, such as the third degree ordeal, was in fact a means to prevent conflicts from spiraling out by forcing feuding parties to come to an agreement, mediated by the church. Since neither party would want to put their limbs into boiling cauldrons, the ordeal instead of actually being used to determine guilt, was meant to expedite the finding of a compromise and solution between feuding parties that could spiral out of control. The ordeal then was more of a threat than an actual institution, and as the church accumulated power and influence over the course of the Middle Ages, it was substituted by alternative methods of determining guilt. Some of these were the same witness calling practices that I mentioned earlier, others were call backs to Roman law and the extreme emphasis on torture in the investigative process (though with exceptions where the Church was concerned).