St. Johns is definitely part of the "First Coast"; along with Duval, Clay, and Nasssau and other areas. You sometimes see it as far as Flagler County.
Like most of Florida's "Coast" regions, both "First Coast" and "Historic Coast" originate in marketing campaigns. Most people don't realize it, but "First Coast" was specifically created and promoted by the William Cook Ad Agency in 1983. Jacksonville and other towns and localities had had individual nicknames before ("Bold New City of the South"; "the Beaches", "Ancient City" etc) but the Jacksonville Chamber of Commerce wanted to develop a brand for the wider region that wouldn't obscure the local brands.
These kinds of regional brands, in particular emphasizing "Coasts", have been employed all through Florida for decades. They rarely last very long and typically have limited public buy-in, but First Coast really took off. For one thing, it's credible and unique - this really was the "First Coast" of European contact in the continental US, and it's the "First Coast" visitors reach when they come to Florida. Additionally, there really was a sense of regional unity that hadn't found its expression. So it stuck and, the First Coast is now one of the best known "regions" of Florida and it's showing no signs of waning.
"Historic Coast" is the St. Augustine, Ponte Vedra, & The Beaches Visitors and Convention Bureau attempting to build a countywide identity. It's a good start, but I really doubt it will last. It's redundant to the much better known "First Coast", and a lot less distinctive. And "Historic" only really characterizes St. Augustine, not Ponte Vedra. It may serve its purpose for this ad cycle, but I really doubt there will be a "Historic Coast News" station thirty years down the road.