FYI...
If you add prompt=select_account as one of the query attributes in the URL on this line of code, Microsoft will ask the user which of their accounts they want to login to instead of assuming.
https://github.com/GeyserMC/global_api/blame/28fd7a60bda23f7dafbdf5571bd85b83194f9e42/assets/js/page/online.js#L5
Making that change means you no longer need the complication of the second paragraph or the logout button this page: https://link.geysermc.org/method/online
:)