lol you're giving me way too much credit, I'm just a guy who likes to run his mouth a bit more than he should :P. Also I'm a bit hypersensitive to tone, and your original post came off a bit more annoyed/frustrated than your comments (which have been very controlled and well thought out), so forgive me if I came off a bit too confrontational in that first comment.
I guess too if you want to be really precise about it, "support" is both a role and a type of champion. In team builder, you only specify your location (ie. Top, Jungle, etc.) and your type of champion (ie. Marksman, Assassin, etc), not your role in the meta (ie. ADC, Support, etc). It's a bit confusing since the word "support" has two meanings in LoL, one where it describes the role in the meta (ie. a Support goes bot lane with an ADC), and also a type of champion (ie. Soraka is a Support/Mage, Ashe is a Marksman/Support). There's always been a conflict with Riot's categorization of champions and the champions' role in the meta, largely because the two are trying to accomplish different goals (for example, Marksman and ADC aren't truly synonymous; Master Yi is technically an Attack Damage Carry, but not a Marksman, because he has no ranged attacks, even though he is a carry champion that's based on attack damage. On the flipside, Teemo is listed as a Marksman, but deals mostly magic damage and usually builds AP, and so is typically not considered an ADC). Riot is creating categories that are meant to help players understand the general playstyle of certain champions as they are designed, where as the meta game is more meant to figure out how those champions can actually fit into a team and what their role in that team is. This is why you have champions like Annie, who Riot lists only as a Mage, being played in the Support role. Since the parameters that Riot has built into team builder only specify the type of champion, technically listing Annie bot as a Support is incorrect, since she's a Mage champion.
Where this gets confusing is that you can mix a "role of a champion" with the "type of a champion"; Riot doesn't believe that their labeling system is gospel for how a champion should be played, only as general guidelines on how the champion is generally played or what playstyle their kit supports, which is why they give you the ability to specify the type of champion you're playing in order to allow flexibility in trying new playstyles; in that way, the designation of "marksman" or "support" isn't truly a description of the champion you're playing as per Riot's labels, more how you're playing them (which makes it sound a lot more like the role you're playing them in). In a way, the "support" role is a bit more flexible than the other categories (you can't play Garen as a marksman, it just doesn't make sense since he's not ranged) since there are a lot of ways a non-support champion as per Riot (like Annie) can function in the role of a support in the larger metagame.
Long story short, listing Annie as a bot lane mage isn't technically incorrect; there are about three types of champions that are typically played in the support role: a mage (like Annie), a tank (like Alistar), and a support (like Soraka). I think your frustration results from an equivocation between the role of support and the champion type of support, the latter of which is actually what Riot makes available in the champion search terms.
Realistically, most players (including myself) generally make these equivocations simply for ease of use, but I think they're important distinctions to keep in the back of your mind.