You should organize your site in a way that makes sense to users, supports their goals, and helps your business/client meet its goals. Rather than trying to "win" this argument, I'd encourage you to get on the same page with your coworkers...Part of doing that is making sure you're using the same language to mean the same things. But because that's probably not a satisfying answer:
Your site's structure should be represented in the navigation insofar as users need to find things on your site. It's up to you (the designers of the site) to decide whether it would be better to include all pages in both navigations, or if some pages don't need to be in the main navigation but need to be in the footer menu, or...the options are really limitless.
Many sites have multiple navigation systems in both the header and the footer: take this site you're on, for example. In the very top header (as of writing) they have things like my inbox and achievements, a menu to switch between StackExchange sites, and a help menu. In the closest thing to a traditional "main nav" that they have, there are links to different slices of the site's main content type: questions. There are also not one but two menus in the footer, with a more expansive list of places around the site (but probably not exhaustive). Take a look at some sites that compare to yours. Try to find something that you also have on your site.
What does their navigation do well? What about it is hard to use? We all have biases when it comes to these things, and that's why usability testing (or user research in general) is so important—it helps your team frame the conversation around what works for users rather than what's the "right" thing to do.
I'd highly recommend checking out Abby Covert's How to Make Sense of Any Mess for a pragmatic, high-level primer for working on information architecture (the specialty that handles navigation and organization of sites/apps/whatever).
