Minecraft – How to make Minecraft PE on iOS forget the xboxlive account so I can start with a new account

This is a quirk that you sometimes get on the App Store. Usually, you can just press buy again, and after you enter your password, it tells you something like:

You have already brought this product and will not be charged again.

However, if you don't want to risk it, you can either: check the "Purchased" tab, anything you download of the Purchased tab should be free, as you own it, or, you could download your purchases onto iTunes, and then sync them on to your device.

If it doesn't have cloud saves, you'll have to start from scratch. There aren't really any ways to back up your data. It could be possible that restoring your device to six months ago when you had it installed might have the relevant save data, but I can't promise that, and it would be awfully inconvenient anyways.