Summing up, it seems like "action API" and "api.php" are the two contenders.
"api.php" is least likely to be confused with anything (only its own entry point file). But as a name it's somewhat awkward.
"action API" might be confused with the Action class and its subclasses, although that doesn't seem like a big deal.
As for the rest:
Just "API" is already causing confusion. Although it'll certainly continue to be used in many contexts.
"MediaWiki API", "Web API", and "MediaWiki web API" are liable to be confused with the proposed REST API, which is also supposed to be web-accessible and will theoretically part of MediaWiki (even though I'd guess it's probably going to be implemented as an -oid). "MediaWiki web APIs" may well grow to encompass the api.php action API, the REST API, and maybe even stuff like Parsoid.
"MediaWiki API" and "Core API" are liable to be confused with the various hooks and PHP classes used by extensions.
"JSON API" wouldn't be accurate for well into the future, and would likely be confused with other JSON-returning APIs such as Parsoid and maybe REST.
"Classic API" makes it sound like there's a full replacement.
All the code name suggestions would be making things less clear, not more. If it had started out with a code name there would be historical inertia, but using a code name now would just be silly.