I will continue to be a stickler on this one. While there is not an authoritative source, all of these interpretations strike me as retroactive. That doesn't mean they have not become widely known, even the one that everybody agrees started as a mistake.
When I say they are retroactive, I mean that the phrase "once in a blue moon" goes back to the 16th century. And for 300 years it did not refer to any lunar phenomenon. It meant something extremely rare (or that never happens.)
Then various people decided they should like to make up a story to describe some sort of rare, or even non particularly rare lunar phenomenon. That includes the volcanic dust making the moon appear blue. That is not a blue moon, that is something somebody heard of and said, "let's call that a blue moon." Same for the other instances with the 3rd of 4 full moons, or the 2nd in a month. The term came first, and the explanation was a retcon. The original reason that a blue moon meant a rare event is lost, or it simply was because the moon is generally not blue. These other meanings conjure up a false story of somebody seeing a rare event and saying, "This only happens as often as 4 full moons in a season" or "This only happens as often as air pollution turning the moon blue." None of those happened.
However, if you want to pretend the retcons are real, you may. But know they are retcons. One can announce that there is a blue moon at the end of this month, but that is not a statement that has any meaning or denotes anything of astronomical significance. Some of the other moons do have some meaning. The harvest moon gave some extra light to work on the harvest. Ditto Hunter's Moon. The others that have names are generally quite obscure. Strangely, of late I have been getting annoyed by certain press getting overexcited as all sorts of astronomical events of various provenance. Supermoons, micromoons, named full moons, minor meteor showers, non-close planetary alignments, very dim comets, etc. And perhaps the most bothersome are the blue moons.
Now I love astronomy, so why am I bothered at the mainstream press shouting the praises of the sky? Because I think hype, in the end, just does more harm than good, and leads to disappointment. Telling people to go out to watch a meteor shower with a ZHR of 20 (and that's only after local midnight in dark skies) just is going to lead them to disappointment. Saying to watch the "supermoon" which is 1% bigger than average and you couldn't possibly notice it, is again going to just disappoint. All these things will disappoint. The blue moon is the most nothing of all; just an invention about a modest coincidence of 12-month and lunar calendar.
Promote the cool stuff, the eclipses (even partial though make it clear how very different those are from total,) and naked eye comets, and even this year's new moon Perseids (though even they aren't that spectacular unless you get people out after 1am in dark skies.) A Bortle 3 milky way. Close alignments of planets you can actually notice. Occultations of planets or bright stars. Transits of the sun. Things people can actually be impressed by. Plus of course views of great objects in telescopes and binos. These are the things that might recruit a new astronomer.