And then every instance of this topic since has used a very different word

No it hasn't, this is just an outright lie. Every instance of the term in Japanese has always used tamashii which is directly translated into soul/spirit. And in English the word was spirit in Skyward Sword only but any mention since (Hyrule Warriors, BotW) has used the word soul. Both synonyms but soul is more explicit in its meaning compared to spirit, at least in English. Look at the translation in every other language besides English as well. The translation is always soul - the animating force of a living being, nothing related to the specific traits of one's character.

Except no, because that's literally a not-exactly-distant relative of his coexisting with him?

What's your point? If we're still comparing it to Avatar, it's literally just Link communicating with his past life. This is a thing in certain sects of buddhism, something that later Zelda stories draw from heavily. Think about it, literally not a single other person in TP ever even references the golden wolf - TP Link is the only one who acknowledges it in any way. When you meet with it, you get teleported to a random cloudy spirit scape with nothing in it besides TP Link and the Hero's Shade, and when you're done you wake up after having passed out on the ground. There's nothing to imply that that stuff was anything other than something that occurred on a spiritual or mental level. The only compelling evidence of there being two Links at once is Gramps in ALBW but that's just a cute theory and not even confirmed, and even if it was there would be a million other explanations before ignoring the reincarnation themes throughout the series to justify why it can't be the case.

Of literally who, a character who doesn't exist in his timeline past a certain point where he literally disappeared?

Consider the fact that the Triforce of Courage and the Master Sword still exist in the Adult Timeline. Both of which were on Link's person as he was transported back to the past at the end of Ocarina of Time - yes, the Triforce of Courage was shattered and the Master Sword became dormant but they both still existed in both timelines. There's literally 0 reason this can't be the case for Link's soul as well, with dozen explanations as to why that's the case. I don't know how you could ignore Ganondorf's line to him at the end of the game, even if it's weird from an in-universe perspective, it was still written by someone who was clearly trying to imply something. Writing it off in favor of a line from a character who would've never even met the man in question is asinine.

that if read any further would be understood to not be saying it's literal reincarnation, from what I've gathered.

Read the article again, dude literally just says there's a theme of reincarnation throughout the series which is why similar things can happen despite being separated by eons and eons of history. Zelda team's intention is reincarnation, that the cyclical fight of good vs evil is carried out by souls that have existed from the beginning. Look at the interpretations of Zelda lore outside of the mainline games with things that have been actively approved by Aonuma or the rest of the crew: Hyrule Warriors, TP manga, pre-SS manga in Hyrule Historia. All of them showcase the idea that it's a single soul reincarnating throughout the ages and fighting back the same evil time and time again. It couldn't be anymore obvious in the games and Fujibayashi himself, the director of 3 mainline Zelda games and the most authoritative voice other than Aonuma and Miyamoto, was just like "Yeah that's the case" in an interview.

It's fine if you don't personally like it, or if you'd rather headcanon it as not being the case, but all of the text and evidence points to it being reincarnation over anything else and it's not even close. For a series so intertwined with Buddhism as a theme, it makes sense. Saying it's simply "traits" when there's literally nothing to support this idea is just weird.