According to the DMG, there are plenty of celestial elves hanging around Arborea. I think "you are really a celestial, fey, or fiend, but everyone thinks you are humanoid" (or more officially Extraplanar Origin) sounds like a good background with the "roll a d8" table for why you are slumming it up in the mortal realms. Especially if 8. is "Lolth in her demon lord mode took a fancy to you, and you saw what happened to the last guy or girl she hooked up with, so time to visit one of those quaint mortal realms (for your health)." I figure demon lord Lolth is probably amused by your escape attempt (and think of the fun she will have when she comes to get you; spider-based curses for everyone in a 1000-mile radius), and no one actually escapes goddess mode Lolth.
The stories about Corellon and Lolth were emerging earlier than when 4e standardized the Feywild and Shadowfell. How these two planes relate to the material and astral planes seems a bit awkward in corner cases. In various senses, the Elf is all four planes: fey, material, astral, even shadow Shadar-kai.
Regarding the earlier terms "outer planes", I call all of it "astral". These are "astral dominions" in the "astral sea", and all of it is in the astral plane. So a technical jargon "astral" can include both celestial and fiend, as well as whatever the modrons and slaadi are.
I want a return of the 4e jargon "planar origin", distinct from "creature type".
I am still less sure how to type the Elf. I went back and changed the original post from "fey touched" to "fey ancestry", even tho it seems narratively less accurate. It is more familiar for 5e jargon. I might revisit it if I have a more solid sense how elven origin relates to taxonomy.
If I look at the terms in a certain way, I think I can find a "Humanoid of Fey Ancestry" sensical.
Different aspects of the elf matter to different players.
For example, Tolkien fans (similarly Greyhawk) might emphasize the material plane, and how for them the elves are more like a human ethnicity with a hint of magic.
For British folkbelief fans, the elf is a humanlike fairy, thus they might emphasize the D&D fey plane. Likewise, 4e fans might champion the elves and eladrin of the Feywild.
The astral domains are the setting for Corellon and Lolth. Corellon is a celestial and Lolth eventually becomes a fiend. Corellon is a shapechanger. The elves parthenogenically grow into shapes from the blood of Corellon, thus are celestial as well, according this narrative. The fans of this setting might emphasize the astral plane for the elves.
I am a fan of Norse folkbelief, where the elf is an animistic nature being. To translate this concept into D&D, they are normal features of nature in the material plane, literally elementals who are native to the material plane. They are the fire of sunlight with its air (Norse) and the earth of fertile soil with its bodies of water (Celtic).
I have made my peace with elves being fey. The idea is, the Feywild is one of the echoes of the material plane. When the normal features of nature animistically project their mind/spirit into a human form (or any other form), these immaterial forms happen in the fey plane of existence. This works well enough in 5e. The only difference is, I emphasize how the Feywild overlaps the material plane, in the same way that the ethereal plane and the Shadowfell do. From the Feywild one sees the material plane, albeit it appears vibrantly. Each fey creature is the projection of some specific feature of nature in the material plane. Animism is all about the physical world, there is nowhere else. So animistic beings journey the "border" of the material plane.
For example, each naiad in the fey plane is a specific wellspring gurgling water somewhere in the material plane. Each hamadryad is a specific tree somewhere. The dryad is more a manifestation of some forest collectively. Similarly, the elf is more sunlight collectively, or fertile soil collectively. In any case, this one-to-one connection between material nature and a fey being is important.
(How to reconcile this animistic view with Corellon and Lolth? The two figures exist in some of my settings. I treat them as the "Adam and Eve" sotospeak of the elf family. The two are elves albeit now at epic levels. Perhaps Lolth led a faction of elves from the astral to the material. In the aftermath of a conflict, Corellon sent almost all shapeshifting elves from the astral plane to the material plane, where they lost their ability to shapechange. They became specific features of normal nature, including sunlight and soil. From there, these natural features can animistically project their thoughts into humanoid forms within the fey plane. From the fey plane, their magic visits other planes of existence. Some made their remote projections materialize into humanoids of flesh and blood in the material plane. In other words, elves are avatars of features of nature and maintain a vital connection to nature.)
Elves can become creatures of any plane of existence. If the technical term "Humanoid creature type" means simultaneously both astral and material, perhaps this creature type makes sense for elves. (They are astral thought become material objects of nature.) By contrast the elves with Corellon are astral only, specifically celestial, and never became humanoid.
(The Fey Ancestry might make sense if every "humanoid elf" ultimately derives from the fey plane − from an animistic manifestation of some material natural feature to there. The nonhumanoid "celestial elf" has nothing to do with this entanglement with materiality. Perhaps the humanoid elves who are still in the fey plane have a more direct connection with the material element they project from.)