I came to Spuffy fandom during the summer between S6 and S7, and in absence of having seen a good chunk of the show (for context: S2 had only just been released on DVD, streaming services were nonexistent, and FX played reruns multiple times a day) I devoured fic and episode summaries. As far as I recall, my introduction to claiming was the story Out of My Head by AJ Hofacre. It was one of the first Spuffy fics I can remember bingeing, which puts its original publication date around 2002.
Claiming, I was told at some point, had its origins in the Vampire Masquerade RPG. I have RECENTLY been told this is wrong, but having never played it, I cannot say for sure. What I can tell you is that vampire romance of the early 00s played with the idea of fated mates long before that term became a marketing tool. I started college in 2003, and was introduced to vampire romance, which I ALSO started to devour in conjunction with Spuffy fic. Christine Feehan was one of the earliest authors I read (incidentally, I do not recommend her books of this era as the men are terrible) but fated mates was prevalent in her Carpathian vampire world. Finding THE ONE. She wasn't the only one; Katie MacAlister had something similar (her fated mates were called Beloveds, iirc), and I did intensely enjoy her Dark Ones series. Whether or not it's aged well, I cannot say. There were others -- many, many others. It was very common in the popular vampire or paranormal romances released during that time. I don't think this popularity has ever gone away, either, just taken new forms over the years.
The first fic I wrote with a claim would've been after Cupidity, so probably getting into 2003-2004 territory. Whether I saw more claims in fanfic before I decided to do it, I don't know, but I have to imagine my decision to start writing them was a combo of having seen it done already in AJ's work and the amount of PNR I was devouring that included similar concepts.
Anyway, I started writing claims, and my fics became known for it (for better or worse). The rules were always up to the author to determine (at least, that was the way I approached it). I'm not sure how personally responsible I am for claims being popularized in 00s-10s Spuffy fic, though I know I wrote them to the point of redundancy at a time where I was especially prolific (tip on being prolific: be a student with a part-time job still living at home rather than an adult with stupid adult shit to do). It could be that other authors were reading the same books I was reading and developing the same ideas; I remember much of fandom being around my age at the time (which was YOOOOOOUNG) so we might've jointly gone on this journey together. But also, this is how fandom culture evolves; someone sees a thing, likes it, and recreates it. I know I am heavily associated with claims (I am mentioned in fanlore and everything!) but again, not sure how much of that is how history has decided to remember it or how much is true. I just know I liked 'em and wrote 'em until I took my fandom break in the early '10s.