To be honest y'all, OH&S is right here. The fact that they said 'a certain third season is coming out... and the fact that it was said right after that stuff, does it mean...?' means absolutely nothing. I've seen them do this before. Technically, it was said at the end of the stage performance after other stuff too. It happened after all of the stuff in the show.
This is a trick that they can use to pretend that they were telling the truth when they were teasing fans that Index III is the thing coming out, but in reality they were talking about a different season III. That way they can trick people until people's emotions about it die down enough so that they just won't care anymore or feel strongly enough to do anything about it. I've seen worse shamelessness coming from these sorts of announcements before.
I don't think anything related to Index III has been confirmed, and I also think that the ToAru facebook page isn't the best source to trust since everyone is biased to hope to Kamisama that season III will happen and so using the bad habit they get from reading Index they look at every little thing that happens and they're like this is the sign that it'll happen, which is obviously silly.
More to the point, all the news sites want views and so saying that season III might happen and is talking about Index is just click bait so that they can get ad revenue. I bet most of those articles don't even believe that's talking about Index III, but they know that they can make more money that way.
That all aside, I'm hoping for Index III for other reasons. I think that it's possible that the biggest obstacle to Index III is that they need to find an opportunity where involved in the project is free at the same time to be able to work on season III. My opinion is that this has the biggest issue this entire time because it seems as if after the movie, they have faith in being able to turn a profit by making anime stuff, and so the most likely setback is being unable to recruit everyone that they need for the project.
At the same time, the series is just way harder to animate starting from season III on. This is partially because of how many details they left out in seasons I and II and partially because the plot becomes that much more complex and because as Accelerator and Shiage get a larger role, Touma's diminishes, I mean he doesn't even appear in Battle Royale. It's hard for them to replicate all of the POV things that happen in the novels with the side-charas and it's hard for them to talk about the philosophy since they're then just wasting animation on words. I really would like to think that these are the reasons why we haven't gotten season 3. If they did manage to animate it as well as (or better than the novels) then I could imagine very good relative success with it, but that seems really hard to do.
Anyway, I'm mostly being patient because I feel like I have nothing to lose by waiting longer for it and that since the novels are better anyway, I'm not losing anything by hoping it happens anyway since it's not the end of the world if it doesn't happen...