Mail and Web not synchronizing

[RobinS]

I guess I haven't set this up correctly.

"Remove copy from server after retrieving a message." is checked.

Below that "When removed from Inbox" is chosen.

So I thought that meant when I move an email from the Inbox to Trash the same thing would happen on the Mail server. But when I checked it today, everything was in the server Trash - nothing was in the server Inbox. Yet in Mail my Inbox has several messages. I want the Mail server to store them as a backup in its Inbox so I always have a copy in case one of these things gets corrupted.

What am I doing wrong? I tried IMAP but it didn't work - or its too complicated to set up. I'd love to get this to work with POP - so my email Inbox on the server doesn't fill up.
Otherwise I have to log on frequently as my server Inbox fills up fast. I hope that makes sense.