Is the circulating story about the Companion whose cloak was stolen by a thief, and the Prophet ordered the cutting off of his hand, authentic? And does it imply that forgiveness for the thief is permissible before he reaches the judge? And does stealing something in a crowded place, such as public transportation or a market, where the victim sees the thief, not necessitate cutting off the hand?

The story mentioned in the question is true and serves as evidence that forgiveness regarding hudud (prescribed punishments) and pardoning them occurs before the matter is referred to the ruler or judge. Once a case is brought before them, the judgment becomes a right of the Sacred Law, not of the individual.

The difference between theft (sariqa) and embezzlement (ikhtilas): Theft is taking money secretly, while embezzlement is taking something openly and fleeing, or by deception and inattention on the part of its owner.

Both are great injustices for which Allah has promised severe punishment in the Hereafter for their perpetrators.

As for the worldly punishment: The hand of the thief is amputated, whereas the hand of the embezzler is not amputated according to the majority of jurists. Rather, he is subject to ta'zir (discretionary punishment) and punished in a way that deters him, such as imprisonment and flogging, due to the difference in the nature of each act in terms of the possibility of guarding against it.