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About the logon URL, you can add your company branding to the Sign In page, then you will could use "https://outlook.com/yourdomain.com"; to access the custom logon page.
Above information could help you customize your Office 365.
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This is what I call a "consolation prize". I am looking into my options now but it's because abundantly clear that microsoft has no regard for it's clients boundaries and has no intention of even providing methods of white labeling. Furthermore, it appears that putting everyone on the same authentication page is showing it's faults in implementation since as of this morning ALL of Microsoft 365 is having issues.
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Personally, Exchange online have many difference with Exchange on-premises, it’s not an Exchange on-premise which managed by Microsoft cloud, it is a standalone version of Exchange. You don’t need to make too many changes to your Exchange online.
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This wasn't the case just a couple years ago. BPOS and Hosted Exchange such as E1 were and had been seperate products but it appears that Microsoft is getting even greedyer and is pulling everything and everyone into "Microsoft 365" like a black hole. Furthermore, i in fact had begun looking into having my own exchange server and since the current version of exchange is last years 2019, I went in search for Exchange 2021 and it appears that Microsoft is doing away with Exchange mostly with the exception of only Enterprise class corporations. There is no beta nor does it look like there will be a trial version available for anyone. I think if i recall it requires installing a whole VM lab to get a trial going. I don't remember but i didn't dig much since it appears that Microsoft has also done away with the open licensing portal where I used to be able to be able to download upcoming versions of software thanks to software assurance and that was the final straw, I'm looking at alternative and same goes for all my clients. Microsoft has no business sticking it's name out past it's customers.
Me personally, i wouldn't even want to show that Microsoft was hosting my email at all! Microsoft has been hacked beyond belief at levels that may very well have compromised every computer running a Microsoft application it. So it's kind of even embarrassing to show I'm hosted and not at the very least running my own private exchange servers.