I've thought more about it, and I'm going to go ahead and declare a particular linguistic nuance:

When it is an adjective, people are more likely to say "twenty." When it is a noun, they say "two thousand."

Consider:

"Back in two thousand fourteen"
"The twenty-fourteen Olympic games"

"When Donald Trump was elected in two thousand sixteen"
"The twenty-sixteen election"

Yeahbut,

In 1814 [eighteen fourteen] we took a little trip
Along with Colonel Jackson down the mighty Mississip …

ETA: and we're gonna party like it's 1999...
……...and the 2001 attack on the WTC...

Bad country music is no way to define anything.

This is good.

I was gonna say, I always say "two thousand nineteen" in reference to what I can't believe is still happening.
"It's two thousand nineteen--people are still [whatever the thing is]??"

All numbers prior to 2000 are irrelevant--no one ever said one-thousand-x-hundred anything. Likewise, once we hit the 2100s, it'll for sure go back to "twenty one" all the time.

2001-2009 are their own category because "twenty oh one" sounds dumb. But I maintain that years like "two thousand fifty-four" will stick around to some degree all the way to "two thousand ninety nine."

All it takes is one really catchy song, that popularizes saying the year as twenty-something in the lyrics, to change the way everyone says it for at least a decade. We'll see what pop culture says.

That's not very catchy.