What does 1 and 2 do?
Both allow you to override the default initialization for an array.
InitArray is specifically initializing the vars array with no parameters, I believe it will assume 0 as the default parameter.
Array is not specifically initializing the array, so it is falling back to a default initialization case.
Why they don't zero initialize like this: bool vars[2]={};?
You could do this, this is just another option which encapsulates the bool array in a class to allow you to provide other functionality if you wish.
What is the purpose of arrayInit() : and array()?
If you want default functionality, there is no need to encapsulate the array in its own class. Encapsulation allows you to encapsulate a type to provide different functionality from the default, you could go on to add methods for addition, subtraction, or anything you can think up and have it perform the methods in the way that you specify.
and what is it called so I could search for it?
Good question; Encapsulation, class initialization, array initialization.
http://www.cplusplus.com/doc/tutorial/classes/