Cold storage refers to security ie keys off-line usually with a device such as signing device aka hardware wallet. Your coins are never out of circulation they are able to be spent at any time without any permissions necessary no gate keeper.
Hot storage refers to a wallet where keys are stored on an online device such as a mobile wallet. Less secure, ok for smaller amounts and spending.
Both coins are in circulation, cold or hot just refers to key storage. Granted if you are saving for long term it would be foolish to store in a hot wallet as it’s less secure so better as a spending wallet.
The only coins “out of circulation” are coins that are permanently lost and cannot be recovered.
You can spend cold storage Bitcoin at any point you wish to do so without consulting or getting permission from anyone.
I hope that helps your understanding. Bitcoin is a deep rabbit hole.