You may have experienced this: when you want to go to the Sign-in page of the Staff Microsoft 365 (M365) Outlook Web App (OWA, the Webmail for M365), you found that you were redirected to Student/Alumni M365 OWA Sign-in page, or vice versa. It is because browser cache (temporary memory) remembers previous status/role of OWA access.
There are two simple ways to solve it. If method (a) doesn't help, you may try method (b) as well.
(a) Quit all existing browser windows and then open it again. Or open another kind of browser (e.g. Chrome or IE) instead.
(b) Restart computer.