Microsoft's email services are still a naming mess. Live, Outlook and Hotmail all seem to be interconnected, and I don't really know what differentiates them anymore.
Except outlook is the app you install on android or iOS to access your Outlook.com personal email, which is really hotmail, but not to be confused with outlook web access, which is the web interface to exchange’s mail. There’s also the outlook that’s part of office 365 for windows, which is “the business one”, except it is also in the office 365 personal edition, which you are not allowed to use for business, and that outlook is a completely different app from the outlook in office 365 for mac.
But the mail client on windows is not outlook, that’s just windows mail.
The Windows mail client was called Windows Live Mail for a while, and the mail service was called Windows Live Hotmail. The gaming platform, video editor and photo viewer were also all inexplicably Windows Live.
It surprises me that they didn't release 'Windows Live Office' in that era.
Here's the fun part: if I type in Hotmail.com into my browser, I'm redirected to a page titled 'Outlook' that's on a subdomain of Live.com (outlook.live.com)
But if I type in Live.com, I'm redirected to the same page that says 'Outlook' but is actually outlook.live.com
So Hotmail is Outlook but Outlook is Live.com but Live.com itself is Outlook