Greentext

I never finished the Silmarillion, but from a synopsis I read, he was defeated first, then the ring was cut from his hand.

I’ve only read the first book, but I don’t actually recall them mentioning explicitly that his finger got cut off in the way the movies did it

Yes, but the rest is very different.

The last alliance spend years preparing and forging elven weapons for the human/elf army, then they marched south along the Anduin and kicked major ass in the Battle of Dagorlad. The elves almost ruined it, because the sindarin didn't want to listen to the Noldorin Gil-galad, so they charged and got their faces punched in in the Dead Marshes (well, that's what they called em after the battle anyway). A WHOLE lot more named characters died.

The Last Alliance breached the black gate, "simply" marched into Mordor, and starting laying siege to Barad-dur for seven whole years. Anarion, son of Elendil and brother of Isildur, gets crushed under a rock at some point. Sauron eventually comes out and fights Gil-Galad. Elendil, Cirdan and Elrond join in and they fight Sauron at full power. Sauron turns in Gil-galad into a pointy-eared torch and smashes Elendil to bits. But Sauron is fought down too, and only THEN does Isildur cut off the ring, banishing his spirit once again. Isildur basically kills him while he's down, not in the middle of the fight and definitely not in the middle of the battle of Dagorlad. There is also no dramatic scene at Mount Doom in the books, but both Elrond and Cirdan have a talk with Isildur about destroying the ring. Isildur takes it as payment for the death of his brother and father, and dies just two years later.

So the war of the last alliance killed the elven kings of Greenwood and Lothlorion (meaning Thranduil becomes king). Elendil, king of the Dunedain, dies and is succeeded by Isildur. There is no more king of Noldor, since Gil-galad didn't have kids.