Seed Phrases

So I am newish to crypto and I was having a discussion with some friends regarding wallets and seed phrases. Please correct me if I am wrong but its my understanding that at the exchange level (Coinbase, Crypto(.)com etc.) your seed phrase and private keys are held by the exchange and therefore you do not own the crypto you purchased. It's all my understanding that with a hot wallet (Exodus, Trust etc.) you have access to your seed phrase and private keys so you have complete access to your purchased crypto. My question is with cold wallets (Ledge, Trezor, etc.) and how are they different than a hot wallet. If you can use your seed phrase to gain access to your digital wallet from your cold wallet device doesn't that hashed seed phrase need to be stored online? A friend of mine involved with this conversation asked if all you need is the seed phrase to gain access to a wallet then a cold wallet is no more secure than a hot wallet. Another question I have is do cold wallets store your crypto coins on them, if so how does that get back to the block chain to record the movement of said crypto. Thanks all sorry for the newbie questions I am about to deep dive this so I can be clear when I tell them why a cold wallet is more secure than a hot wallet.