It's "in the cloud", aka the blockchain

I am not talking about the website blockchain.info (it's confusing they use the same term).

It's in the blockchain that bitcoin uses, part of public record

Only you have the "private key" to do anything

I'm honetly not sure how blockchain.info doesn't ahve your private key. They need it to do anything.
I think they store, on there servers, some peice of data, that when combined with your password, turns into the
private key