In the medical literature, "intoxication" often means a milder form of "poisoning," or they are just used as synonyms. So, you can usually find "vitamin D intoxication", and "arsenic poisoning". But when water is contaminated with arsenic, it may cause (only) "arsenic intoxication."
"Alcohol intoxication" usually means being heavily, but not necessary dangerously, affected by alcohol or, if you want, being out of control, and the term "alcohol poisoning" is reserved for a heavier, life-threatening condition, with greatly affected consciousness.
"Water intoxication" is an intentionally incorrect, fancy term to emphasize the life-threatening effect of drinking excessive amounts of water and it may be presented in the medical literature as "the so called water intoxication."
"Water poisoning" is sometimes used for "water intoxication" as above, but also for the infections with waterborne microbes, so like "food poisoning" or even for the poisoning with mercury or other poisons in water.
What I've written above is how the terms are used and there is not necessary much logic in it. But if I insist in explaining terms from the exact meaning of the "poison" and "toxin:"
To my understanding a
poison is what
Merriam-Webster says:
1: a substance that
through its chemical action usually kills, injures, or impairs an organism
2: a substance that
inhibits the activity of another substance or the course of a reaction or process <a catalyst poison>
Toxin, in biology means a poison produced by a living organism.
So, if "intoxication" comes from the "toxin" as defined above than "water intoxication" is incorrect, since there are no toxins involved. And "water poisoning" is also incorrect because a shift of electrolytes caused by excessive water drinking is not already a chemical reaction where two substances would react and produce other substances.
"Alcohol intoxication" and "alcohol poisoning" are also technically incorrect, because, in these cases, alcohol effects are due to release of certain substances in the brain and not due to any chemical reactions, which would make alcohol a poison.