The implementations currently are just SQLite hooked into the browser, they aren't different in any real way.
It is a bit circular, but generally something gets proposed as a standard, multiple browsers will implement it, then it becomes a finalized standard.
In this case it was proposed, the standard says "your implementation needs to do what this other implementation does" (and no other information is given about the SQL system), 2 browsers (really 1 engine at the time, webkit) used SQLite directly, and it never went anywhere else.
I guess my point isn't that there's not a "spec", but that the spec that there is doesn't say anything about the SQL part of WebSQL, it just says what the interface to the SQL engine looks like, and then mandates that the SQL engine act, look, and work like the one in a specific version of SQLite.