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I think your answer is quite interesting, and quite involved for math stack exchange. Presuming even that it is right, I don't know why the question should be closed.
@jorl David hampings I think you're right, I didn't think much on that last "appendix" axiom, I wanted to edit it even before Iv seen you comment but I didn't want to change up the question too much.
I wouldn't say that this question should require some charity or that it is evolving. Only change I made is some standard and trivial parts of the metric definition.
Oh so you mean that requiring that a subest has pairs of points of any arbitrary real distance between them still doesn't imply it's maximal. In that case sure OK, it can be further required for it to be maximal.
