So, to wade in and try to explain!
The initiator is an ingredient that will start the polymerisation process but it will not start this until it is 'triggered'. This is where the catalyst comes in as this is the 'trigger'.
The catalyst is not altered in the chemical process, it is just the trigger. The initiator is, however, part of the process and is altered in some way and becomes part of the polymerised plastic. benzoyl peroxide, for example, is completely used up very early on in the process.
All of the 'systems' need both a catalyst and initiator to successfully polymerise in a controlled way. The relationship between these two is very carefully balanced formula in sophisticated products. A very good reason for not mixing brands!
As Geeg has said, there is a certain amount of confusion when using 'initiator' as a noun and a chemical term and 'initiate' as a verb in normal vocabulary (e.g. initiate conversation, i.e start conversation)
With reference to my book the text explains this relatively complicated process and how it differs in the various systems. (There are many aspects of my book that go above and beyond what the typical technician needs but that is my approach! Know and understand as much as possible as problems can then be solved by deduction for those that like this level of understanding. Essential in my belief ) However, the Glossary has complicated my explanation! This and the index is written by other individuals. For some reason benzoyl peroxide is listed as a catalyst! I missed it so no excuse!!
The relevant explanation is in the main body of the book.