But is a phase pistol the same thing as a phaser?
I see no reason it couldn't be. That's exactly how terminology tends to evolve over time.
Much less a laser (as used in the first couple TOS shows)?
While the "laser" props from "The Cage" were reused in the second pilot, they were referred to there as phasers. Roddenberry decided that it had been a mistake to call them lasers in the first place, because real lasers didn't behave anything like that and he feared people would complain about the inaccuracy (meaning that he either underestimated the audience's suspension of disbelief or overestimated their education). If he'd done a new story set in the time of "The Cage" or earlier, I have no doubt that he would've called the weapons phasers instead of lasers.
but the Star Trek Encyclopedia confirms that phasers and lasers are different types of weapons.
It doesn't "confirm" anything, because it's not a canonical source, just an interpretation like any other tie-in. It contains errors (like referring to Nella Daren as "Neela") and assertions that were overwritten by later canon.
Evidence from the TV shows indicates that the creators wanted EM-33s and phase weapons to be different technologies. Are we just insisting that phase weapons and phasers are the same thing because of the similar names?
You're the one talking about what the creators wanted. Why would they have even
called them "phase weapons" if they hadn't intended them to be at least a direct ancestor of phasers?
There are distinct differences, like the number of settings, for example. If I recall correctly, the technobabble behind their operations and functions is also different.
Yes, and TNG phasers had 16 settings while TOS phasers had 4-5. And modern cars have a lot more gears than early cars. And modern TVs can get hundreds of times as many channels as early TVs. What, did you expect the technology to remain absolutely unchanging for all time?
Also, along the same lines, wouldn't Worf's line prove that the any 22nd-century weaponry was not a phaser? (Worf may not be a historian, but he is a weapons expert, per his job and hobbies and the definitive nature of the statement doesn't really leave any wiggle room).
A single unsupported anecdotal statement can never be taken to "prove" anything. I'll never understand the tendency to assume that every single line of dialogue in a story must be accepted as absolute, unquestionable fact. People are fallible. They misstate things, they misremember things, they misunderstand things. That's why anecdotal testimony is never acceptable as scientific evidence, why anecdotal historical accounts must always be verified by alternative sources, why hearsay is not admissible in court, etc. Why should fictional characters be any less fallible than real people? The only thing that Worf's statement "proves" is that he
believes there were no phasers in the 22nd century.
Besides, it's fiction. Retcons happen, inconsistencies crop up. As the video showed, there are plenty of contradictions in Trek. There is no "proof," because it's all made up and later writers can disregard what earlier writers assumed if the story demands it.
Also, for what it's worth, the novel Academy: Collision Course describes phasers as being a fusion of phase weapon and laser technology (ch. 19, p 134 - 135, paperback ed.). (The Shatnerverse novels aren't quite in continuity with the rest of the novels -- although Cast No Shadow implies that they are -- but that version could be correct.)
Again, it's fiction, not history. We're not obligated to acknowledge the conjectures and interpretations of earlier novels; we can do so if we choose, but it's not mandatory.
Besides, it's possible to read too much into terminology. Even if phase weapons and phasers are different, maybe phasers were called that because the term was already coined as a nickname for phase weapons. The digital watch on my wrist is powered by a completely different technology than the watch my grandfather would've carried in his pocket, but they're both called watches.