A Wide Area Network is only only network, not two
You're losing me there. Where one geographically dispersed network stops, and the other begins is arbitrary anyway - is all determined by your perspective.

most people may be unfamiliar with this kind of usage because many work with only the Internet and not with *independent internets. But I assure you, they exist.

I get where your coming from Brandon, but. I've worked in the area a while and I ain't heard one person term an expansive network, 'an internet'(..admittedly I work in a country where English isn't spoken) - but you've got to admit, it's more than a tad confusing. Guess my sketchy(sic) definition of the Internet was o.k. though - or would you like to offer your own?
Let me conclude by saying that internet and Internet mean one and the same thing to me - if a technician/specialist is referring to some communications network he/she is usually going to be speaking on a little more precise terms regardless.