First, a note on terminology since I'm going to deviate from my normal usage to avoid some potential anachronism. I'll be using the Haudenosaunee to refer to the political unit and the Iroquois to refer to the people who formed it. Likewise, the Wendat will be used to refer to the political unit and the Huron will be used for the people. Iroquoian refers to a much larger linguistic and cultural association that includes the Iroquois, the Huron, and many others.

I know that the Iroquois originated from the "Huron stock" as one website liked to put it

This isn't accurate. The Iroquois and the Huron are cultural siblings, rather mother-and-daughter. The recent roots of Iroquoian peoples seem to trace back to the Allegheny Plateau region. There were two notable migrations out of this area by Iroquioan people. The earlier of the two headed south and became the Cherokee at an unknown time. Later, between 650-800 CE (give or take a century), the Northern Iroquoians pushed out in waves heading in various directions. Some went south as well to become the Tuscarora and other southerly Northern Iroquoians. Most headed north to settle around the eastern Great Lakes and through the St. Lawrence Valley.

By the time the Iroquois founded the Haudenosaunee, the Huron were their own people, as is evident in one of the most common variants of the foundation narrative in oral history (since it says that the Great Peacemaker came from the Huron).

the Huron split off into it's own nation, right? And they didn't join the league, right?

You're right that they didn't join the Haudenosaunee but, as mentioned. they didn't split off.

Which is why the Iroquois nations waged war on them (and their French allies), correct?

We don't really know what the casus belli for the Haudenosaunee-Wendat War was. While the constitution of the Haudenosaunee (the Gayanashagowa--The Great Law of Peace) does permit the invasion of an enemy nation that has refused to join on at least three occasions, this was unlikely the primary motivation. There were economic reasons for expanding Haudenosaunee's hunting territory. There were demographic reasons for bringing in captives for adoption in order to recover some of the loses from the recent epidemics ravaging the region. There were political and military concerns with having a French presence in the area, as the French had attempted to invade the Haudenosaunee previously. Most likely, it was a combination of these factors, rather than a hypothetical rejection of the Great Law, that pushed the Haudenosaunee to invade in force.