I'm confused about POA and POI on red dots. The way I understand it, a Red Dot "scope" type optic requires your centering the dot in the reticle, but a holo type sight does not.
I had a cheap Tasco red dot on my .22 pistol and quickly discovered that I had to keep the dot centered. I've also shot an EoTech on my friends AR and discovered it didn't matter...if the dot was on target, so was the bullet.

So is this the major difference between a "scope" and a "holo" type optic, or do I have something screwed up?

You don't have to keep the dot centered with either the red dot or the holosight.  As long as you are shooting in the parallax-free range on the optic (usually beyond 25-50 yards), the round will go where the dot indicates.  Either the Tasco's glass doesn't correct for parallax or the mechanicals have let loose (most I've encountered cut in and out every few 22LR rounds)

A conventional red dot will have a smaller sweet-spot where you can pick up the dot, as you are looking through a tube.  An EOTech has a very large sweet-spot, as the front and rear glass are much closer together.