A word that simultaneously means caring and not caring?

Solution 1:

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..]

Solution 2:

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