A sending server does not look up the recipient's address. It asks DNS which hosts handle mail for the domain, sorts the answers by priority, and works down the list until one accepts the connection. That is the entire mechanism, and it is why the priority column matters more than it looks — the lowest number is tried first, and higher entries exist as fallbacks.
Resolving every exchange also tells you where mail physically lands, which is often more informative than the hostname. A domain nominally running its own mail can turn out to point at a hosted filtering service, or at infrastructure belonging to a provider it stopped paying years ago.
Why another tool may disagree
DNS answers are cached, and every resolver holds its own copy until the TTL expires. Just after a change, one tool can legitimately show the old records while another shows the new ones. Neither is wrong; they are reading different caches.
MX records say who accepts mail for a domain. To check who is authorised to send as it, use the SPF checker and the DKIM checker.