But could you give an example of an event so improbable that it is נכון to assume that a higher being was involved in it? Because so far the only example I’ve found of that is abiogenesis, and even there one can invoke the theory of multiple universes, so that the probability could in the end still be possible.

Multiple universes (not infinite ones) are not relevant to the discussion. The question is: who created them? Have you ever seen a random universe-generator? This is a "turtles all the way down" type of explanation. And if indeed in your view there is such a generator, then its name is God. Nice to meet you.
I also gave the example of the die, and others along the same lines. For example, if you yourself roll a fair die and suddenly get such a result. Or if you hear a voice from heaven teaching you the theory of quantum gravity, and it turns out that you have indeed solved that complicated problem with no education in physics. In such cases I would definitely consider the possibility that this is a supernatural event.

It seems to me that there is a difference between the implausibility of abiogenesis and the implausibility of a person winning the lottery 10 times in a row, and likewise in the example of rolling a die. In the case of the die roll and the lottery win, we know that in any event there is someone behind the drawing or the roll, so you can point an accusing finger at him and say that he caused it to be this way, whereas in the case of abiogenesis we have no prior proof that any such entity exists.
Also, as far as I know, it is not enough to know about someone that he won the lottery 10 times in a row in order to put him in prison without additional evidence; it simply raises suspicion very sharply.
Are my remarks correct?

In my opinion, they are not correct.
You can point a finger, or you can refrain from doing so. Someone rolled the dice, but who says there is someone responsible for the results? It came out by itself, so why accuse anyone? Therefore there is no principled difference between the cases. From the specialness of the phenomenon one can infer that there is an entity behind it that is responsible for it.

So basically, in a certain sense, the physico-theological argument is just a different formulation of the watchmaker argument we know, right?
Or perhaps the watchmaker argument relies more on subjective feeling, whereas your argument relies on more objective probabilities?

Indeed. I’ve written that more than once. The arguments against Paley’s watchmaker argument are not correct.

I’ll bring here a quote from Prof. Nathan Aviezer’s book that shows a refutation of Paley’s watchmaker argument:
"Snowflakes are extremely beautiful crystals, with a complex and delicate structure. Each one has precise hexagonal symmetry, and each differs from the others. Nevertheless, we all know that there is no need for a designer of snowflakes. The delicate beauty of snowflakes is not produced intentionally, but rather they are formed naturally when suitable weather conditions prevail."
After that he also gives the example of a person who sees letters of the alphabet engraved on trees in a forest, and from this infers that some person wrote them. But in fact this inference is mistaken, since the reason he attributes special status to those letters is his familiarity with the Hebrew language and his past experience with those signs. From nature’s point of view, nothing unusual happened here, and from this one understands that the inference is incorrect, since it rests on a subjective feeling.

He does say there that the refutation does not apply to the origin of the laws of nature themselves, but Paley’s watchmaker argument, as I understand it, does not speak about them but about the products themselves.
What do you think about these claims?
Sorry that it came out in two messages; I just have a bug on my computer that doesn’t let me write too much in one message.

I wrote that too. The proof is from the laws, not from the phenomena.