Hieupt, you are right, the page I referred you to wasn’t detailed enough to help you with your sentence; I apologize. You are correct in saying that your sentence ought to have been “it won’t probably rain” according to this rule, which means that the rule doesn’t apply here, since that would be incorrect.
The problem is not the word “probably”, but the fact that the sentence is negative. “It’ll probably rain” is according to the rule, and it is correct. I have never thought about this matter before, but here is what I propose.
We could add a new rule for those adverbs that normally come in the middle of a sentence: when to the sentence a word (usually “not”) is added that negates the sentence as a whole, the adverb comes before the auxiliary verb. I’d say this only applies strictly if verb and negation are contracted, as in “do not” => “don’t”; if they are not contracted, the adverb may sometimes, though not often, go between verb and negation as expected. I believe this depends on the verb: it should be “it is probably not”, but “it probably does not”.
(I mention the “sentence as a whole” because it doesn’t apply to partial negations, as in “I told her not to come”; here the scope of “not” is only “to come”: the “telling” itself is not negated. This as opposed to negating the entire sentence, such as “I clearly didn’t tell her to come”.)
As to the reason why this is so, I cannot say. I have a hunch that it has something to do with the scope of the negation. If you say “it won’t probably rain”, the word “probably” falls within the scope of the negation “-n’t” because it comes after the negation. It is the same as saying: “it is not true that it will probably rain”. However, what we really wanted to say is: “it is probably not true that it will rain”; here, “probably” comes before “not” and thus does not fall within its scope.
I can think of a dialogue:
A: “It will probably rain in Spain tomorrow.”
B: “It won’t probably rain: it will certainly rain! The weather forecast says: 100% chance of rain in Spain.”
Here person B specifically wants to deny the truth of the word “probably”: that is why he puts it in this unusual order. Note that the rules of the placement of adverbs only point to the most ‘normal’ order; in special cases, much greater variation in placement can be found.