Wasn't sure if I should have posted this in the 1980's or 1990's board since it's 2 years of each, but many have discussed that the early 90's were more of an extension of the 80's rather than the 90's beginning in the late 80's. Whenever we think 90's we think more core 90's a bit later on in the decade. Anyhow, here are how each year was pivotal:

1988 - New Jack Swing started to get big as well as pop rap like "It Takes Two" by Rob Base and DJ EZ Rock. For New Kids on the Block and Milli Vanilli, its debatable whether you consider those 1988 or 1989, and depends on where you live (the latter was popular in Europe from '88 since they are from Germany). This is also Reagan's last full year in office, and George Bush Sr. was elected president.

1989 - George Bush Sr. was sworn in office. Many popular Early 90's TV shows premiered like Saved by the Bell, Family Matters, Baywatch, Arsenio Hall Show, and The Simpsons (if you count the Christmas episode). Also Dance House songs like Technotronic's "Pump Up the Jam" debuted, but that one's debatable whether it was associated with 1989 or 1990. The Sega Genesis and Nintendo Gameboy are released in the US/NA.

1990 - Of course the first numerical year of the 90's, but what about culturally? It was the beginning of the regular seasons of The Simpsons (not counting the Christmas episode), and also premier of The Fresh Prince of Bel-Air and In Living Color. Dance House because more popular like Technotronic's "Get Up (Before the Night is Over)" and Black Box's "Everybody, Everybody".

1991- Nirvana releases "Nevermind" which marks the official start of grunge, though grunge is more of a core 90's element. The Super Nintendo is released in the US/NA.

By 1992 we were well established in the early 90's culture and that is the first year that is a contender with 1993, 1994, etc. in shifting from the early 90's culture to the mid/core 90's with the election of Bill Clinton.

I'm going to say 1990. I was there. And I remember the cultural 80s to 90s shift happening that year. By 1991, culture felt more 90s.

1989, for reasons mentioned by the OP and more, even though the first four-five months were still very late '80s. Four of the most iconic '80s shows, Miami Vice, Moonlighting, Dynasty, and Family Ties ended their runs that May. Popular '80s game shows like Sale of the Century, Super Password, Definition, and Card Sharks also handed out their final consolation prizes. Arsenio Hall premiered in January, but nobody at the time saw it as a major threat to Johnny Carson. It seemed like from summer onwards, the real '80s vibe that had dominated for the past seven years had turned the corner towards a new era. Black Box may have gone big with "Everybody Everybody" in 1990, but their real breakout was "Ride on Time" in '89, especially in "soccer" nations.  Nickelodeon's Hey Dude premiered that summer and stayed in reruns for much of the '90s and is often seen as part of the network's golden age. Then the Berlin Wall came tumbling down and Czechoslovakia's communist government crumbled at the end of the year, and the Cold War was in its epilogue phase. Nirvana dropped "Bleach" that year, even if they were still labeled as an alternative band. Disney's The Little Mermaid also started The Disney Renaissance. Sure, 1990 and 1991 still had '80s influences as we know and the '90s as a whole had a lot of the '80s infrastructure intact through much of the decade, but you were a dork if you were a high schooler carrying around Duran Duran albums or a Max Headroom shirt.

That said, there is a big difference between the "Bush '90s" and the "Clinton '90s" in that the former feels distinctively late 20th century, while the Clinton years are more transitional late 20th/early 21st century with things like the Internet and mobile phones starting their mainstream usage, and the origins of "woke" culture (as "political correctness") becoming gradually more widespread.

Yes, 1991 had a shift that was nearly as big as 1989, but it was more of a shift that was pointing towards the CORE '90s, which includes the last bit of the early '90s, the mid-'90s, and the non-Y2K late '90s.