Well that's true - but challenge her with the fact that you're not actually flying to CHC or to PER (using your theoretical as an example) so how can she say they are a stopover using her definition of "flying to somewhere" when you're not actually flying to that somewhere?!

I honestly don't know how to help you further as I can't believe you are getting the runaround on such a basic, simple rule interpretation that's been a part of the :-: RTW rules since year dot.
Both ends of the ground segment are categorically, without doubt, 100% one stopover under number of stopover rules, but all 4 cities are counted as a city stopover under city stopover rules.

There have been several post threads/posts with helpful links or email addresses to agents that have provided knowledgeable and superior service for RTW fares. Maybe a search for them will help find one that's in your neck of the woods?
Cheers.