Somebody managed to trick the exchange into signing a message saying "move this amount of money from this address to this other address" (signed by the private key associated with the first address, with the second address being owned by the attacker) and submitting it to be added to the blockchain by a miner.

The blockchain is the only record of who owns what money, and transactions on it can't be reverted. Therefore, the attacker stole money by causing a transaction that shouldn't have happened.

Bitcoins were stolen, not a sequence of bits. It's not much different than if you steal USD from my electronic bank account. Bits are obviously involved in the process, but the USD in a bank account is not "just a sequence of bits." No physical property was stolen, but most of us can agree that this qualifies as theft.

This is more a case of fraud than burglary. The attackers tricked the exchange into sending then tons of money.

Burglary traditional implies physical entry of a perpetrator into physical building or space. I assume it would not apply to "breaking into" a computer or network in most modern legal systems. Regardless, this would almost certainly be considered larceny, as well as computer fraud.