The concept design is taken from FFT, which originally called them lancers. Also there was a time period when FF Fans would refer to the class as Lancers, because of the ever popular Legend of Dragoon, as not to confuse fandoms.
You are correct that in subsequent releases of FF games that included the class they have hence referred to them as Dragoons, including remakes of FFT.
North America didn't see another class named dragoon until FF9, and the Playstation release of FF5. Most FF Fans had their start with FF6 or FF7. Those who got to play FF1 or FF4 are historically considered hipsters.