The "Biden Rule" wasn't some kind of formal Senate rule, but rather was merely referring to the course of action that Biden himself suggested in 1992 that you mentioned, namely, not taking up a nomination during the election season.
The name, of course, was used by Republicans just to point out the hypocrisy of the administration (and Senate Democrats) of demanding that Obama's nominee be considered immediately when the Vice President himself (a long-time Senate Democrat) had made a floor speech arguing for the opposite when the parties were switched, even hypothetically before any actual opening came up on the court.
As a PolitiFact article on this subject from 2016 notes regarding this situation and Biden's own words from 1992,
In the case of Obama's nomination of Garland, Democrats have argued that the Supreme Court seat should be filled immediately because the court needs a deciding vote.
Biden in his 1992 speech addressed that issue, saying that some people "may fret that this approach would leave the Court with only eight members for some time. But as I see it, Mr. President, the cost of such a result, the need to reargue three or four cases that will divide the justices four to four are quite minor compared to the cost that a nominee, the president, the senate, and the nation would have to pay for what would assuredly be a bitter fight, no matter how good a person is nominated by the President, if that nomination were to take place in the next several weeks."
As far as use of the term "rule," while there was no formal rule, Biden's speech in 1992 did assert that the President waiting until after the election season to nominate a candidate in such a circumstance was an established (but not legally required) precedent:
As a result, it is my view that if a Supreme Court Justice resigns tomorrow, or within the next several weeks, or resigns at the end of the summer, President Bush should consider following the practice of a majority of his predecessors and not — and not — name a nominee until after the November election is completed.
PolitiFact does note that "Biden's floor speech was on June 25, 1992, more than three months later in the election cycle than it is now," with 'now' in the quote being about the time that Republicans brought up Biden's speech. However, Biden's speech said,
Instead, it would be our pragmatic conclusion that once the political season is under way, and it is, action on a Supreme Court nomination must be put off until after the election campaign is over.
President Obama nominated Garland on March 16, 2016. To say that the Presidential election season was "underway" by then would be quite an understatement. The first primary election (the Iowa caucuses) took place on Feb 1. Campaigning starts several months before that. By the time Obama nominated Garland, not only was the Presidential election season well underway, but 26 states and two territories had already voted in the Democratic primary and 30 states plus several territories and the District of Columbia had already voted in the Republican primary. The United States has 50 states, so more than half of them had already conducted their primary elections in both parties by the time of Garland's nomination.
So, the timing of Garland's nomination was very clearly within the "political season" timeframe Biden had mentioned for avoiding court nominations in his 1992 speech.
As PolitiFact also notes, however, Biden only (expressly) argued for delaying the nomination of a candidate until after the election season, not explicitly until after the next President took office.
However, the political reasoning for this is pretty obvious. By the end of the 102nd Congress, the Senate had 58 Democrats and only 42 Republicans. If a candidate weren't even nominated, let alone vetted by the Senate before the election, then the majority party in the Senate (Democrats at that time) would have a major upper-hand, even if the President nominated someone the day after the election was over.
In such a scenario, one of two things could play out:
President Bush (a Republican) was re-elected. Senate Democrats have lost nothing and can just proceed with the nomination hearings as usual after the election.
Bill Clinton (a Democrat) was elected President. They can now trivially stall the nomination proceedings for a couple of months under the guise of "thoroughly vetting" the candidate. Or even just drag it out for a month and a half, hold a vote, and reject the nominee without enough time for another nomination to be considered. Bush's nomination expires at the end of his term and Clinton fills the seat. The ruse is now obvious to most voters, but the next Senate election isn't for 2 more years and, thus, is much less likely to be negatively affected for them by this behavior than an ongoing election would be.
Of course, in 2016, the shoe was on the other foot and was real rather than merely hypothetical like in 1992. Had Hillary Clinton won the election in 2016, I have little doubt that the Senate Republicans would have decided to "graciously" give Democrats what they asked for and voted to confirm Garland (by far the most moderate of Obama's nominees) in the lame-duck session rather than letting Clinton nominate a more liberal candidate. And this would been doubly true if Republicans had lost control of the Senate in the election (though they didn't.)
The only difference is that, of course, President Obama did not follow the "recommendation" that Biden made for President Bush and wait until after the election was over to nominate someone (because, obviously, that would have been playing into the hands of Senate Republicans, just as Biden was wanting Bush to do for Senate Democrats in 1992.) So, Senate Republicans did face criticism for employing this obviously-political move during an election season, though it didn't end up having enough impact for them to lose their control of the Senate.
Is this tactic all completely politically motivated? Absolutely. Is it unsavory? I suppose that's a matter of opinion, but I'd say so. Is it a formal rule of any sort? Obviously not (though Biden did make the claim in '92 that it was a precedent.) Do both parties do it when it favors them? Yep, at least if they think they can get away with it.