Why can we draw a more precise conclusion when we choose a lower accepted-risk in this hypothesis-testing setting, which seems contradictory? We want to know if 100 integer values (in a vector X) are following a Poisson $P(\lambda=2)$ distribution, which is our $H_0$ hypothesis.
Let's say the observed value of the $\chi^2$ with 5 degrees of freedom is $C=7.5$.

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*With an accepted risk of $\alpha = 20\%$, the threshold for $\chi^2_5$ is $7.29 < C$. Thus we reject the hypothesis $H_0$. Conclusion: we don't know anything about the distribution of the vector $X$, except that it's probably not a Poisson of param. 2.


*With an accepted risk of $\alpha = 1\%$, the threshold for $\chi^2_5$ is $15.09 > C$. Thus we don't reject the hypothesis $H_0$. Conclusion: we now know that a good candidate for the distribution of $X$ is a Poisson distribution of parameter $2$.
How is it possible, in this $\chi^2$-goodness-of-fit setting, to:

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*not know anything when we accept a high risk

*have a more precise idea when we decide to take a much lower risk!

This seems contradictory with the intuition: if we want a very low risk (1%), we should not be able to draw any conclusion at all here... And if we accept a higher risk (20%), we should be able to draw a conclusion.
Why is it the contrary here?
Note: I have read many questions/answers about Fisher / Neyman-Pearson, such as When to use Fisher and Neyman-Pearson framework? and others etc. but I think it would be useful to understand this precise example before being able to understand the whole theoretical setting.

Your question seems to be based on a mistaken premise.
Specifically: "Conclusion: we now know that a good candidate for the distribution of X is a Poisson distribution of parameter 2." is not a reasonable conclusion.
It may be a fairly poor candidate but perhaps you have little power to reject the null.
Failure to reject a null does not imply it's true (only that you lack sufficient evidence to conclude that it's false).
Failure to reject under a more stringent test even less so.
Note that the specific "risk" you're talking about should be made explicit -- it's the risk of incorrectly rejecting the null; that is, it's the chance -- when the null is true -- of rejecting it.
The higher you push that risk (making your rejection region larger) the better your chance of also rejecting a false null (i.e. the higher your power).
You can increase the rejection rate (both when correct and when incorrect) by having a larger rejection region and decrease it by having a smaller one. They move together.
You are (incorrectly) making a strong conclusion when you fail to reject, when the ability to say anything comes when you reject (i.e. that it's not consistent with a Poisson(2)).
It's a bit like a court case (at least in most English-speaking jurisdictions) -- if there's insufficient evidence then you can't convict, but that doesn't mean that the defendant was in fact innocent (for all that the law subsequently treats them as if they were). Absence of evidence is not the same thing as evidence of absence.
Goodness of fit tests don't tell you what you have, only (sometimes) that the data are not consistent with the thing under the null.
