And that's only ever happened with Grover Cleveland and Grover Cleveland (they're two people). In fact no defeated incumbent has ever been renominated since the two-term limit was installed.
No, you are forgetting Teddy Roosevelt (whom Trump closely resembles) -- he was renominated from his Bull Moore party. Also the mathematically famous mnemonic e = 2.7 (Andrew Jackson) (Andrew Jackson). He served two terms: 1828 and 1832 after losing to a surely stolen election "won" in 1824 by John Quincy Adams. Jackson won the popular vote but not the Electoral College (heard that one before?) and Calhoun threw his support behind Adams
in exchange for Secretary of State; this was then and now known as The Corrupt Bargain, because it was. These cases don't meet your exact terms with Cleveland, but they are close: so three examples, and Trump can make the fourth, if he lives so long in health.
The post clearly says "since the two-term limit was installed" but just because you bring him up, TR wasn't a defeated incumbent when he was passed over by the Republican convention in 1912 even though he had the commanding lead in the primaries. His "Bull Moose" party nominated him, not "re"-nominated. It was the first and only time it did that.
Almost worked too. The Republican came in third, making that the last time a "third party" had any bearing on the election outcome.
But what 1912 also shows is that a political party can and does nominate whoever it wants to nominate, primaries or no primaries, which in turn means the same party could have done the same thing --- a former First Lady's mantra of "just say no" --- in 2016. But they declined to take that option, and now they're paying the price in Chaos.
Now let's do Jackson, the only POTUS besides Washington to get the office without a political party. Yes he did pull both more popular and electoral votes than Quincy in 1824 but did not get enough EVs to trip the Presidency, so as the Constitution prescribes it went to the morass of Congress, which chose Quincy in the infamous "corrupt bargain". But the fact remains he didn't win, and was not sworn in as a result of, the 1824 election, ergo he was not an incumbent in 1828.
Cleveland (and Cleveland too) was (were) nominated in 1892 after losing as an incumbent in 1888, but that too was before the 22nd Amendment. But I'm not clear on whether the defeated 1888 Cleveland was the first Grover Cleveland or the second identical Grover Cleveland.
Speaking of Cleveland though, the Prez who interrupted his term and made him two people, the memorable Benjamin Harrison, was the only POTUS candidate to lose the popular vote TWICE, until now. Not counting Quincy (who also ran in 1820 and lost) because the popular vote wasn't really a thing, certainly not nationally, until after the Civil War.
Finally, back up to your first line, "TR whom Rump closely resembles". Fair point --- bombastic egomaniac from New York who bellows a lot. The difference is that TR made adjustments to his ego while Rump, even as we speak, is punching the walls in the Oval Office, screaming in the anguish he created for himself, unable or unwilling to think his way out of it.
