Could somebody please explain the difference between omitted variable
bias and endogeniety bias?

With these biases in mind, please critique the regression starting on
page 11 going to page 12. (section 3 to 3.1) of
http://www.opelcreative.com/slaverypaper.pdf

Namely, given that variation in slavery could be because of low income
(ie poor people wanted slaves badly), how does white level control for
this.

I assume that omitted variable bias is where you omit statistically
significant varibale in your regression, whereas endogeniety bias is
when your dependent varibale and independent varibale move in the same
direction causing multicollinearity problem.