A full backup or home ?
 

If I want to move my website from one host to another (from Lunarpages in my case) what kind of a backup do I need to download : a full account...

The best approach here is the hosting provider itself. You should check with Lunar pages and see what options you have from them. https://imagicon.info/cat/6-5/1.gif

Hello,

If you are using cPanel control panel at your old host then you can ask them to provide a Complete account backup which will include site data, email accounts, emails and databases too.

You should learn the techniques of cloud migration so that you will be able to know that what do you need and what procedure you will need to follow if you are transferring your data from one host to another.

Good luck!!