Blockchain.com, formerly Blockchain.info. It is a light client wallet, not a full node client. If talking about full client, you do not have to talk about Blockchain.com wallet, it has nothing to do with that at all, it is a light client.

Bitcoin Core, formerly called Bitcoin QT is a full client wallet that you can use to run your own node after downloading the full blcockchain or you run a pruned node.

Bitcoin is stored on the blockchain, an open ledger which is available to anyone. You will need to download the blockchain (which is where bitcoin are stored) to run full node.