It's a "name" used to describe something that does not exist as rule, and barely as an inconsistent and "ex-post" rationale. Wikipedia's entry makes three valid points:
1) It was not a rule or even informal agreement.
2) Both parties invoke the "name" of the rationale when it suits them, and both flip flop when circumstances dictate.
3) The specifics of what happened to Fortas are so vastly different, and the origins of the 'rule' so dubious, as to suggest the rationale (and Thurmond mythology) was invented sometime later (the WP suggest 1980, not 1968).
Yes, the opposite party does tend, but not always, to slow down appointments during a final year before an election. However, the Fortas nomination to Chief Justice was successfully filibustered after his hearing (the cloture vote not close to being achieved) AND the for all the reasons given in both the wiki entries on both Thurmond and Fortas NONE of those reasons included the basis of the later invocation of a Thurmond rule.
The death of a supreme court justice during the year of an election, with different party's controlling the Senate and Administration, was unprecedented in modern S.C. history. (The last case being somewhat similar to vacancy in the 19th century). And to the degree it was similar to any case, it was similar to Eisenhower's appointment, with the permission and support of the Democratic Senate, of a Democrat to the Court.
Had Obama followed this precedent and nominated a Republican to the Senate majority's liking, the "controversy" would have been unnecessary. And, by the way, we would have not had to hear of anther 'rule' you left out - "the Biden rule".