Despite the fact that "under", "below", "beneath", and another option you've neglected, "underneath", overlap significantly in meaning, they are in practice not distributed in the same way -- "under" is far more frequent than the others, and "underneath" far less frequent, even outside of the context of water. In Spacial Prepositions and Metaphor, Frank Boers proposes that this is because while each of these prepositions have slightly different characteristics that make some of them suitable in some contexts but not others (or provide for differences in meaning -- consider the difference between "He was wearing nothing underneath his shirt" and "He was wearing nothing below his shirt"), "under" is treated as sort of a "vague default", so that it can be used in a more general sense and thus using one of the alternatives seems marked in a way that using "under" does not.
As for what these specific differences are, there are many resources that address them. Cambridge dictionary specifically instructs users not to use "below the sea" and to instead use "under the sea", because while "under" can be used when something is covering or touching something else, "below" is generally not used in this way. Their entry for below also specifically says that it is used when there is no contact between the people or things involved. English grammar.org also says that "under" is used when one thing is covered by something else (it directly cites "under the water" vs. "below the water" here) and that "below" is "mainly used in cases when an object is not directly under another". As you can see, this is far from isolated to some native speaker(s) on this StackExchange -- it's pretty well-discussed that "below" is less suitable than "under" in these contexts.
These definitions gel with my personal (native speaker) intuitions as well -- if someone said that something were "below the water", I would assume that they meant that thing was buried in the earth that rests beneath the entire body of water, because "below the water" carries implications that it is not touching the water, which not the case if it's submerged in it. Based on the comments on this question, it seems this intuition is the same for JakeRobb.
The information I can find doesn't address "beneath" much except in non-literal cases and to describe it as much more formal/literary than "below" and "under". I think it patterns like a more formal version of "below", but I don't really have any evidence to back that up.
As for why you can find examples of "below the water" even if it's generally perceived as unidiomatic on a more widespread basis, there are probably a few factors at play. One is that if you're referring specifically to the surface of the water, "below" suddenly becomes much more idiomatic, because this is usually describing something that is not touching the water's surface. "Below the surface of the water" is actually more natural-sounding, to me, than "under the surface of the water" (though mileage may vary on that). It's possible that in some contexts or for some speakers, "water" can be used with its surface being most salient, leading to "below the water" feeling more natural. It's also possible people sometimes make deliberately unconventional choices here because they're more marked and thus feel more novel/less cliche -- this would be my explanation for titles like "Life Below Water" and more poetic uses of the 'wrong' preposition. Further, not every text is written by a native speaker, and these prepositions are so similar in meaning and overlap often enough that even highly proficient non-natives could easily choose "below" in a context where most native speakers wouldn't. Most texts aren't being proofread so strictly as to change them unless it leads to actual difficulty understanding the text for native speakers (rare in this case, as the surrounding context usually makes it clear that "submerged in water" is what is meant).