Pre-European Inhabitants of the Allegheny/Pittsburgh Area

Please excuse me if this is the wrong forum. I'm writing an English essay to advocate for a Native American monument in Pittsburgh (Specifically Boyce College). I have been reading about Native American settlements and history in Pennsylvania but it is very confusing and too broad a scope for my paper. It's also hard for me to find concise sources on the Internet (maybe sluthing isn't adequate). I wanted to know whether this brief and abbreviated summary of the early history of Allegheny County/Pittsburgh area is right please?

The earliest recorded inhabitants are the Aniyvwiya (also known as the Cherokee) according to the Lenni-Lenape (Delaware). The name Allegheny means the river of the Alligewi (or Talligewi, translation is unclear)—the Lenni-Lenape name for the Aniyvwiya.

The Lenni-Lenape and the ancestors of the Haudwnosaunee (Iroquois) colonized the region and displaced the Aniyvwiya in a war long before European contact. The Haudwnosaunee subsequently drove the Lenni-Lenape to the present day New Jersey/Delaware in a war. A splinter group of the Haudwnosaunee's ancestors, the Erie, inhabitanted the area until European contact.

Eventually the Haudwnosaunee destroyed the Erie to control their territory for the fur trade with France. The Seneca, a member nation of the Haudwnosaunee Confederation, dominated the area until they were defeated by English colonists in the French-and-Indian War.

Is this accurate? The Cherokee.org seems sure that they were the Alligewi, but the other sources I read seem uncertain to make that leap. Once source claims the Lenni-Lenape's history goes back to the migration from Asia, but I find that questionable (maybe it's my own prejudice)?