No, Philistine (and all the variants) comes from a Greek word for "uncouth" and is a word for the ancient Philistines given by their neighbours; it's unknown what the Philistines called themselves. The Philistines weren't the Phoenicians, they were more recent invaders (possibly some of the "Sea People"). For one, the Philistines were Aegean and the Phoenicians were Semitic. The Philistines also disappeared (either killed or assimilated) while the Phoenicians spawned Carthage (the ones in the Levant probably just assimilated over time as many powers controlled the area after them).

It only became the name for the land after the Bar Kockba revolt, the Romans named it such specifically to spite the Jews. And then it stuck when various powers controlled the land over time (Romans/East Romans aka. Byzantines, Caliphate, Ottomans, British).