The definition of cold or hot wallets depends on where the private keys are located and if they have ever been used to sign and send transactions to the blockchain.
What is a cold wallet?
A "cold wallet" holds private keys associated to addresses on the blockchain where the private key is on paper or on a hardware wallet, but has never been used to sign a transaction.
This is considered the highest level of security in the blockchain industry because once you sign and send a transaction to a blockchain there is some traceability and information now on the public domain that could conceivably, however unlikely, be used to access your crypto assets.
Although improbable, this marginal risk is sufficient to warrant keeping large amounts of crypto in this type of wallet, however inconvenient they may be to manage.
When a cold wallet is used to sign a transaction, then it becomes a "hot wallet", even if it's on a hardware wallet or paper.
If a private key is on a software wallet, it is "hot" even if it has never been used to sign a transaction as it is connected to the internet inside your machine or phone, which is a less secure environment.
What is a hot wallet?
A hot wallet is where the private keys associated to addresses on the blockchain have been used to sign one or more transactions. It doesn't matter if it's in a hardware wallet or software wallet.
Any keys, used or not, on software wallets are considered hot wallets because they are readily available to sign transactions and are in a less secure environment than hardware wallets or cold wallets.