The bitcoin network is an independent and decentralised network. Other networks or websites don't run in the bitcoin network neither does the bitcoin network run in other networks. If I may, you are actually likely refering to a centralised exchange that allows you deposit in a a said address you claim is yours without having the keys (since they literally control your keys).

When you deposit in that address, they wait for the transaction to be confirmed on the bitcoin network or lightening network as the case may be and then the allow it reflect in your balance on their CEX website or wallet.

As for you idea of setting up an exchange or a website that accepts bitcoin payments or deposits, I suggest you contact a good developer and team .