Usually what websites do is to build the burn transaction for you. In reality, there will not be a difference between burning from using a website or using the CLI tool. They will both interact with the SPL Token program to generate the burn instruction.
To burn from your wallets you gonna have to export your wallets. Then use the JSON file in the CLI to burn. Only the wallet can burn its own tokens.
Another solution is using a website like https://soldustvacuum.app/, where a burn instruction is built into the process and it will decrease your supply because it is interacting with the SPL Token program.
burn instruction example
So either export your wallet to JSON and use it on the CLI the same way that you burned the first time, or use a website. The final result will be the same.