When is a cipher considered broken?
Your explanation is "broken in the academically sense" - there is a theoretical way to break the algorithm better than brute force. AES is broken in this way. There's also "broken in the practical sense", which means you can break an cipher in real applications or protocols. AES is still safe in this sense, because you still need too much computation power and even related keys to get the original key.

There are also different attack scenarios. You can have ciphertext-only attacks - the attacker only knows the ciphertext. That's a pretty weak attack scenario and not realistic. What if the attacker knows some of the encrypted plaintext because it has a pattern? That's the known-plaintext attack. There are some more kinds of attacks with even more help for the attacker, but they are harder to do in the real world, but even if an algorithm is broken with one of this harder threats, that doesn't mean that the algorithm can't be used anymore. AES is secure because the attacks still need too much work to get the key.