Saints Row 3 and 4 are both good games. In some ways the later games outshine the second game.
But one dimension shines in the second game which I found to be gradually fading in the later ones, the story. It depends on your preferences. The later games embraced a more crazy plot in place of a little more coherent one.
However, the third and fourth games are fun in their own right. The mechanics are solid as is the engaging gameplay. I played the third one with idolninja's modpack, though it is abandoned in beta for reasons, and is not suggested by the author. (it is still good, at least I found so, but use it at your own risk :) )

I care a great deal about the story and characters (as well as all the little details that make the world feel alive) so that makes a big difference to me.

So I've come to understand, and that's why I want to play the third game (I'm still undecided about the fourth). :)

Thanks, but I'm hopelessly old-school and never mod the games I play.

It seems you've already bought the games, but I have a few suggestions just in case you're still listening and don't necessarily have to buy from GOG.

The game that I recommend pretty much everywhere and to pretty much everyone is Portal (and its sequel, to a lesser extent). It is a puzzle-platformer, so technically doesn't fit your criteria, but the portal mechanic means that there is a lot more to it that just jumping. There's even a story, and the first game is pretty short (maybe six hours on your first playthrough), so you may find it gripping even if you don't like puzzle and/or platform games otherwise or if your attention span is short.

Then there's The Talos Principle, which, like Portal, is a short-ish first-person puzzle game with some mild platforming and a story. I think this would be especially good for teenagers who think of everything, think big and want to know everything (like I was ages ago). I don't want to spoil the game too much for you in case you want to have a go yourself, but basically The Talos Principle trades a progressing core story for a mostly static background against which it projects philosophical themes (which is not to say that there is no core story, of course).

Are there any unique individuals among them, or are they just stereotypical American teenagers?

Spore! It's a nice game, a great one for younger audience, which is a reason why it's still a popular title among YouTubers. It incorporates many types of gameplays through the various eras of evolution, starting with the cell stage in the "primordial soup" and ending with the space era. It also spur creativity and imagination to customize your creature, buildings and vehicles, and you can also share your creations with the online community.

Hmm, teenagers might like Sims 4, Cities: Skylines or Cook Serve Delicious 2.