There are plenty of AP champions that are not mages. Most of them are AP Assassins (like Akali, Diana, Fizz), some of them are AP Tanks (Cho'gath, Ammumu). Mages is the word we use for spell casters, champions that their primary damage is through spells AND they need to use their spells frequently (as opposed to a one combo burst, which would be an assassin).
Teemo just happens to be the only AP Marksmen, meaning his damage comes from auto-attacks. You /can/ itemize Teemo for burst so that his Q will do a lot of damage (That's why his second classification is actually Assassin, because he can burst really hard). But even with a lot of AP he does not majorly rely on casting multiple spells, so he is not a mage.
Corki is closer to secondary mage, because he does rely on spamming his abilities (although he is still only classified as a marskmen).
Lots of people agree that Teemo isn't in the best spot post-rework, but considering how recent his rework is Riot probably won't tweak his kit for a long while. Instead, he'll probably get slight numbers changes. I've seen a lot of people specifically asking for changes to how long his shrooms can last.