I think that “modern” accepted definition of chareidi is not a response to haskalah, it’s a response to the Holocaust. As was noted, the Chazon Ish was the one who really started the Kollel system as it exists today, in response to the loss of so many leaders. I think that the kollel system is the defining characteristic of the modern chareidi movement.
Chassidus has also had a huge impact on the charedi world, with many chumros that chassidim followed being adopted by the chareidi world.
My Rav likes to say that chareidim are those who follow the Torah, and that’s it. He also says that “Modern Orthodox” means those who figure out how to view the modern world through the prism of the Torah. Therefore, he said, the biggest Modern Orthodox rabbonim were R’ Moshe Feinstein, R’ SZ Auerbach, R’ Elyashiv, Chacham Ovadia Yosef, etc. who took the new modern things we have, whether it’s electricity, medical advances, or anything else, and figured out how the halachos we have apply to them.