They look, sound and act exactly like phasers, so they're phasers. I guess we're supposed to assume "phase pistol" was shortened to "phaser" by the time of The Original Series.
Worf's line in "A Matter of Time", like Spock's lines about cloaking devices being a theoretical new technology and 22nd century starship technology in "Balance of Terror", Picard's speech about Klingon/human first contact in "First Contact" and the "lasers" in "The Cage" were simply ignored. Such is television, especially something as long-running as Star Trek. Generations of new writers and producers come along, all with their own ideas about how the Trekverse should work.