There is no universal agreement on how to exactly define what airline and isn't a "legacy" airline.

To some people, it's any airline that follows a traditional business model. In other words, it's any airline that isn't classified as an LCC (Low Cost Carrier, where Cost refers to operating costs, not customer fares). Southwest plus all US-based stand-alone airlines that were created in the last decade or so (Jet Blue, Virgin America, Spirit, Sun Country, etc) are classified as LLCs, and Frontier seems to morphing into an LLC (though they seemed to be a smaller legacy carrier as of a few years ago).

The term "legacy" refers to age (either of the airline or simply of the business model of the airline; ie, the LCC business model didn't exist many decades ago, so any airline that was formed many decades ago and has stayed in the same business model could therefore be considered a "legacy").

It gets confusing, though, if a new stand-alone carrier adopts a legacy business model despite being a new carrier.

IMHO, the people who think Alaska and Hawaiian shouldn't be considered legacy seem to be confusing the term "legacy carrier" with "networked" carrier.

The term "networked" carrier typically refers to a carrier that flies signficantly overseas as well as domestically, interlines baggage, etc, and indeed only AA/US, DL, and UA are left as US-based networked carriers.