Answer

Hello.
This is his well-known responsum about lotteries (I discussed it a bit in my attached article on the Torah portion of Pinchas, in the weekly portion). First, this is talking about Purim. Second, these are not people who gamble regularly, and in Jewish law the ruling follows the view in the Mishnah in chapter 3 of Sanhedrin that the gambler is disqualified only if he has no occupation other than that. And third, he discusses there several kinds of lotteries, where if they are done properly then the result is determined from Above, and if I remember correctly, it seems from what he says there that this is permitted.
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Questioner:
There is a Mishnah in Sabbath chapter 23:
A person may count his guests and his dishes by memory, but not from writing; and one may cast lots with his sons and the members of his household at the table, provided that he does not intend to make a large portion correspond to a small one, משום gambling with dice; and one may cast lots over sacred offerings on a Jewish holiday, but not over portions.

I understand from here that it is forbidden to cast lots even casually. As for the types of lotteries, apparently the Sages did not distinguish between different types of lotteries regarding the prohibition of gambling.
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Rabbi:
See the commentators there, who explain that this is a prohibition of theft because of asmakhta. But in the Sanhedrin passage that I mentioned, the amoraim disagree whether gambling with dice involves asmakhta or not, and in practice Jewish law follows Rav Sheshet that it does not. The Mishnah there, stated anonymously, follows the opinion that there is asmakhta here, but that was not accepted as the halakhic ruling. And in his Commentary on the Mishnah, Maimonides elaborated that according to all opinions there is still a problem here in terms of proper conduct, because it is not fitting to engage in this (at least if it is his main occupation in life, as I wrote there).
Beyond that, I wrote that the distinction between different lotteries is a novelty of the Havot Yair, and the novelty is precisely that we really do not find a distinction between different types of lotteries, as you wrote. Still, you should look at his words there themselves (I didn’t check just now, and I wrote from memory).
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Questioner:
In general, according to Jewish law nowadays, is it permitted to make a monetary bet or wager on an occasional basis?
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Rabbi:
As I wrote, according to Jewish law it is permitted to do this occasionally. The practical difference in the dispute between the amoraim in Sanhedrin is exactly whether the prohibition applies only when this is a person’s main activity (if it is merely an issue of proper conduct and contributing to the functioning of the world), or whether every such act is prohibited (if it is theft). In practice the ruling is that there is no theft here. In Maimonides’ Commentary on the Mishnah there he wrote that doing this regularly is still prohibited, though it is not clear whether he meant an actual prohibition or merely something improper.