I think I can make an educated guess about class width, but anyone else feel free to correct me.
For example, you may be looking at frequency distribution of football players in a school (let's say school has all 12 grades). The classes may be (1st and 2nd grade), (3rd and 4th grade), (5th and 6th grade), ..., (11th and 12th grade). So the classes are not "joined continuously", because the limit for the first class (2nd grade) does not coincide with the limit for the second class (3rd grade). By definition, class width = 3 - 1 for the first class (or 4 - 2 if you subtract the upper limits). It is not class width = 2 - 1, i.e. the difference in the endpoints of the class.
In #2, it is an example. It is NOT another definition. In this example, the classes are joined continuously (like they often are). So actually you could subtract the endpoints of the class to get the class width (since the upper limit of one class "joins continuously"/coincides with the lower limit of the next class).