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can I switch to IMAP, try it, and go back to POP
Maybe/sort of. First, your provider would have to have both services available, or you'd have to have at least one POP provider and one IMAP.
But it's not going to be fully equal swap when/if you switch. The major difference between the two methods is:POP is basically a one-way system.You request your mail, with any one device.
That device downloads it to only that one device.
The message is deleted from the service.
But it is stored on whatever device that downloaded/read it.
IMAP allows you to download/read the message one many different devicesAs many times as you want (until you delete it from the service)
And all your other devices can still read/download it until it finally gets deleted.
You may never actually store the message on your own device(s).
So, once you use POP to view your message, it won't be available to any other device (unless you send it to yourself, again). In other words, 'you can't go home, again.'
OTOH, what you read/download on an IMAP account will still be there if you decide to use POP. So, you could 'go home, again.' At least once for each message. 
Back when we had only one device (like a 8MHz 16K RAM computer) POP was fine. Now, many have a computer, a smart phone and a laptop/iPad and want to have all their messages available on all devices, no matter which they might be using or where they might be on Earth or nearby it.
The above is quite simplified and generalized, of course. Some POP systems can have be set up to act similar to IMAP in some ways, and vice versa.
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do I have POP or IMAP?
You already answered that in your post.
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Apple Mail Preferences says "POP".
Incoming mail server - pop3.live.com
First, I'd fix the iBook so you can send mail. You should have exactly the same outgoing mail server (smtp.live.com) and password as you have on the iMac. Make sure the settings in Accounts->Account Information as well as Advanced are exactly the same in both computers.
If your ISP has IMAP available, that's probably the type you need. As has been stated above, IMAP allows all devices to 'see' the same messages on the server. From Apple's Help "About Account Types: IMAP accounts:QUOTE
With an Internet Message Access Protocol (IMAP) account, incoming and outgoing messages are stored on a mail server until you delete them. When you connect to the mail server, all your mailboxes are synced so that the messages on your computer are the same as those on the server. You can read and manage messages from any device that you use to access your account, and download messages or attachments to work with when you’re not connected to the Internet.
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Although when I SEND mail on the iBook, it seems that email can't be duplicated on my iMac?
I suspect that your messages from the iBook are not being stored on the POP servers, or, you have settings on the iBook telling Mail to erase downloaded messages when they are received. Since any message you send from the iBook will also be received by it, almost immediately, it will not stay on the server for any appreciable time and not be available on any other device. Bt remember that you can get the same kind of behavior with an IMAP account if you tell the server to delete a message when it is received. Those settings are up to the owner of the account and whatever email app they are using. In other words, an IMAP account can be set up to act just like a POP account, if the user really wants it that way.
Have a great trip!