Oh my fozzy this really helped!-I've owned four of these species, and from my experience, here's what I have for you. It's all about the individual and like Stormy said, any bird can be very unpredictable.
Budgies: solid choice. But they are very skittish and a LOT of them are not hand-tame and never will be. They can be playful (down to the individual) but are hard to keep track of, and it's very easy for them to get stuck somewhere while flying or playing. And in these kinds of situations, they panic and it's very difficult to get them. I think a budgie would work for you but...yeah.
Parrotlets: they are absolutely wonderful - I'm sure you can tell by the way I talk about Cricket. But they are living blobs of affection and SASS. They are very smart and can learn a lot of tricks, and are not half as skittish as budgies. They aren't goofy, but they're feisty. It takes a lot to keep up with them. Bottom line: not for you, not yet. Sometime in the future, though, you might be a great match for them.
Sennies: you know my Lilla is a very sweet girl, but many sennies aren't...like her. Most are more playful and outgoing, but not cuddly and cute all the time, and will be unpredictable. And are hard biters (thanks Skarila). Maybe though...? I think you could handle one, as long as you know what you're getting into.
Caiques: yes, I wouldn't recommend them for children, in most situations. Zoe, you are not inexperienced, I know that, but it's not even the bites. Caiques ARE hard biters and very unpredictable. But they're just clownish troublemakers in general. It's extremely difficult to keep up with them, especially when you go to school. I got Nutmeg as a student (older than you though) and it was hard. Amazing, but hard. And while they're what you said you're looking for..."PLAY PLAY PLAY", they're as big a commitment as any large bird. Not saying you couldn't consider getting one, just saying that I think you should wait if you want one.
You did a great job with Tango and Tiki. I'd say another conure or tiel would be an okay choice too.
Thanks a bunch