The term "flag carrier" is anachronistic. It comes from an era where airlines were annoited by their respective governments to "carry the flag".
It means very little, if anything today, especially in North America.
There are numerous Canadian airlines, Transat and Skyservice come to mind, that fly into places the old "flag" carrier doesn't fly. To people in Austria, I would imagine Skyservice is Canada's "Flag" carrier. To people in Exeter, Transat is Canada's "Flag" carrier. Transat would definitely be the "Flag" carrier to Scotland. Who is the "Flag" airline to the Dominican Republic? WestJet? Air Canada? Transat? Skyservice?
It's slightly ironic that Transat, CanJet, Skyservice, Zoom and Sunwing all have Canadian flags on their livery. WJ has the flag and "Proudly Canadian" on it's livery. Other large Canadian based airlines do not carry the flag. Weird, eh?