I am trying to grasp when to create a bootstrap.php file and how to structure it. It looks a lot like an init.php file. Are they the same concept? Or maybe the same exact thing?
In the context of web-applications and PHP, the concepts "bootstrapping" and "application initialization" are more or less the same. The filename doesn't matter.
As for the details of what such an implementation should contain, that is completely dependant on the architecture of the application.