Problem
I don't understand the difference between a 'hot' crypto wallet, and a 'cold' crypto wallet.
Assumptions
My understanding of blockchains is that they use public/private cryptography in order to transact.
A wallet address is a public key.
A private key allows a user to send transactions.
Technically all wallet addresses already exist (defined by a hash function that generates a public/private key pair) and therefore are just waiting for someone to use these keys to start transacting.
Blockchains are essentially a distributed database which have data relating to the balances of cryptocurrencies (among other things)
Question
I've read that 'hot' wallets are connected to the internet and therefore less secure, whilst 'cold' wallets are not connected to the internet and therefore more secure.
I want to know if this is actually fundamentally wrong - because public keys are neither connected or disconnected to the internet - they are generated using an algorithm and then used to sign transactions. When crypto is sent to some wallet address in a 'cold' wallet - the cryptocurrency is not moved off the blockchain, it is just associated with some public/private key pair.
Is my understanding wrong - or is there just a huge misconception about what a cold/hot wallet is.