"separates the calls properly" is purely subjective. My answer: Vicidial separates the calls perfectly already. lol.
If your goal is to track THREE levels of "user" instead of TWO (which vicidial is designed to do), then you will want to find a report that shows at least some of the information you see so it may be rewritten to your needs.
We've written many customized reports designed to determine information based on call history. We've also written many scripts that alter data in the Vicidial lead record to explicitly make a determination (such as "who was the verifier?") and store that information for later reporting purposes.
You can also put this information in a custom field on all the lists to allow the Verifier to "choose himself" during the verification call OR write a start_call or dispo_url script into the Closer campaign that stores the user ID in a special field. Lots of ways to go. All of which are "proper" to those who use them (subjectively, of course).
PS: Good job posting your system specs.