BEAST and CRIME are both chosen-plaintext attacks. They don't rely on the improper MAC composition in TLS CBC.

Lucky13 and POODLE are the chosen-ciphertext attacks.

EDIT: Some more details:

BEAST takes advantage of predictable IVs in SSLv3 and TLS 1.0. In these protocols, the IV for each new record is simply the last block of the previous record. An attacker monitoring traffic on the wire can use this predictability to build an encryption oracle and guess-and-check the contents of ciphertext blocks.

CRIME uses plaintext compression to its advantage. A message with longer common substrings will compress slightly better than one without, and this is reflected in the ciphertext length. An attacker can make adaptively chosen guesses at substrings included in the message to recover, e.g., session cookies.