Back in the days where each country had one home-based airline that flew internationally, flag carrier designation was largely a cerimonial designation involving national pride, etc.
Today, here in the US at least, it refers to any US-registered airline that operates scheduled service to international points. US Part 121, which governs the majority of operations here in the USA, sometimes differentiates between Domestic operations and Flag operations by the virtue of the type of flight-related differences involved.
As far as I can recall (and someone feel free to correct me), if a US Domestic carrier wants to run a charter to an international destination, it doesn't automatically make them a Flag carrier, since it's *scheduled* international service that's the key. Non-standard operations such as offline domestic charters are handled (per the airline's Ops Specs) as a supplemental operation, and I think international charters would also be.
Outside the US, "Flag" carriers are still largely a cerimonial designation, notwithstanding any regulatory differences within the regulations of the home country...
ALL views, opinions expressed are mine ONLY and are NOT representative of those shared by Southwest Airlines Co.