"Cold" is about key exposure, not the coins: the bitcoin sits on-chain either way, but a cold setup means the signing keys live offline where remote attackers cannot reach them. The opposite — a hot wallet — trades that safety for convenience.
The giant cold wallets are public landmarks: the largest address in bitcoin is an exchange cold wallet, and Bitfinex publishes its own — the difference between "attributed" and "confirmed" in our labeling methodology. We watch the biggest ones live and alert on major moves.