5e defines prone, and it aligns with the definition of prone in English. 5e is written in English.
In both 3e and 4e, you are supposed to treat keywords as magic tokens and mostly ignore what they mean in English. In 4e they even gave this advice explicitly in the rules text.
This is not how 5e is written. 5e is, first and foremost, an English text.
Snakes are already prone. Making them "more prone" is nonsense. Applying the prone condition to a creature that is already prone does nothing. Well, if the "snake" had wings, I would probably make it fall, as knocking things prone in 5e also knocks (non-hovering) creatures out of the sky. But its new state (prone) would be the same as the old one (prone).
It is consistent to invent a definition of "Prone" that isn't the English word "prone", but there is no huge balance effect here, nor is there a need to do so. Use common sense, and double check it to ensure it doesn't break the game part of the game.
5e, unlike 4e, doesn't have explicit fluff distinct from rules text. The fact that knocking a creature prone actually makes them prone is part of the rules text, it is not fluff you are supposed to ignore.
Similarly the creature is "snake", not "creature with these game statistics with the word 'snake' taped to it". The 5e rules are some of the rules how to model the snake in the 5e combat engine, it doesn't mean you should ignore the fact the we are talking about a snake.