A website stays on the previous host until the domain's name servers, or at least the address records, point at the new account. On this storefront the sample name servers are ns1.cyberfusionsystem.site and ns2.cyberfusionsystem.site. They are labelled sample until the operator confirms the live delegation.
Change both records at the registrar that holds the domain. Changing only one leaves resolvers split. Allow up to 48 hours. The hosting checker on this site compares the text you paste. It does not query the public DNS, so a green sentence there is not a live lookup.
If the domain was registered in the same order and the registry has already accepted it, the name is not refundable. Moving the name servers does not cancel the registration. The cancellation page explains the separate clocks for hosting and domains.
Mail moves with the same delegation if mail is hosted here. Lower the TTL a day before the change if the registrar allows it, and keep the old mailbox available until the new one receives a test message you sent from outside.