Coinbase and Bitpay are companies that offer payment processing services for other companies, like the website you mention. They manage for them all the things related with the payments in Bitcoin. Usually the business will open an account with them and they will provide tools to accept Bitcoin in the website. They will store the bitcoins for the business, send them where it says or even exchange them for dollars or euros if that is what the business wants.
Bitcoin Core, on the other side, is a wallet. A business can use it to store its bitcoins if it wanted, but it doesn't have anything that will facilitate accepting Bitcoin in a website (except having an address). A good practice for anyone accepting Bitcoin payments through a payment processor is to have them sent to the business local wallet (which could be Bitcoin Core or any other) after the purchase. This way, the business has full control of those coins instead of having them stored by the payment processor.
A user can also use the Bitcoin Core wallet. When trying to purchase from a website that accepts Bitcoin, you will get an address to which you have to send the payment. That you can do from Bitcoin Core or any other wallet you use.
Incidentally, Coinbase also offers something similar to a wallet to anyone who wants it. I say "similar to a wallet" because they have a system in which they keep control of the bitcoins, not the user, so it can't be really called a wallet. It is just a website that lets you manage bitcoins that are tied to the account you have there, but they have the custody of the funds.