Cold usually means it needs multiple physical people to sign from offline devices to move it. Hot wallet usually is automated. Here it looks like the «hackers» found a way to trick enough people to sign this transaction
It could still be cold. "took control of the specific ETH cold wallet" sounds like stealing the physical hardware. Like someone stealing the vault key, or the HDCP master key getting leaked.
They could have gotten the recovery phrase off some paper, then imported it wherever. More likely than guessing the pin on a ledger with a short number of tries before wiping.
The wallet is a smart contract (specifically a gnosis safe), the malicious message they signed transferred ownership of that smart contract wallet to the attacker so they could then do whatever they want with it.