I can't help you with the Outlook settings as I abandoned desktop mail clients 10+ years ago and have zero regrets.  In general, many email providers are either 100% hosted or at least offer that.  Who do you use?  There is a good chance you are actually using a hosted IMAP service which means all the messages are stored on the server.  The difference with this, compared to the "old" days, is that your account has lots of storage available, whereas in the old days your account had limited storage.  This is why you downloaded your messages.