A word that simultaneously means caring and not caring?

You might manage to express that idea with // insouciant which is such a self-concious sort of word that although it means 'devil-may-care' it implies a bit of an act.

Otherwise perhaps a paradox: 'deliberately unperturbed', 'with studied indifference', 'puritanically laid-back'. Mix and match until it resonates.

//Insouciant 1829 careless, indifferent, unconcerned. [Shorter OED gives fr and eng pronunciation for which I don’t have the symbols;; try “in-‘sooss-eant” for eng..]

I agree that it will probably be difficult to find a single word for this. How about "steady half-heartedness" or "courteous ennui"?

You can try indifference or nonchalance.

Also, superficial

Concerned with or comprehending only what is apparent or obvious; not deep or penetrating emotionally or intellectually

Teenagers are notorious for being contradictory.
or
Teenagers have a tendency to be paradoxical. They say one thing, but do another

According to Collins Dictionary:

paradox
1. a seemingly absurd or self-contradictory statement that is or may be true ⇒ religious truths are often expressed in paradox
2. a self-contradictory proposition, such as I always tell lies
3. a person or thing exhibiting apparently contradictory characteristics

Answering this much later for future interest: Nonplussed gets at your question - it means both disconcerted / perplexed and not bothered or suprised (chiefly US) However, the problem with nonplussed is exactly the fact that it has two meanings. It almost always requires a follow up to explain which meaning you meant.

While I'm not certain there is a specific word to sum that up (the suggestions thus far are wonderful,) I would recommend considering something concerning the anxiety-resistance cycle that has become quite common among youths.

There is a great deal of anxiety surrounding all aspects of their lives and pressure to do well, but there is also often a paralysis that comes with this. This can be true even if the individual doesn't show or feel the symptoms readily associated with anxiety.

Check out https://www.life-with-confidence.com/procrastination-can-be-signs-of-anxiety.html it has some excellent descriptors that you may find useful.