A Managed Bean is any been defined in the faces-config file(s). It's exactly what its name says - a POJO managed (created and initialized) by the
JSF framework.
A Backing Bean is a bean that's backing up a JSF view. That is, it's part of an EL expression that references that bean or it's properties on a JSF web page definition.
Since managed beans are created on-demand and the most common demand is from a JSF View, you can consider the two terms to be essentially equivalent.
The secret of how to be miserable is to constantly expect things are going to happen the way that they are "supposed" to happen.
You can have faith, which carries the understanding that you may be disappointed. Then there's being a willfully-blind idiot, which virtually guarantees it.