Autozone says 2 different alternators and advance auto parts says 1. how many different alternators did the 2000 mercury cougar have and how do I know which one my car need?

There are two different Engine used in the Mercury Cougar. The V6 and The 4 cylinder.

Since yours is the V6 that narrows it down to only One. A 130 Amp Vision OE. I am sure they offer a New or Reman(Re-manufactured). But that is up to you if you want a new one.

Part# 7775 or N7775 from Vision $150 from local parts sore
or
Part# 210-5196 from Denso $120 from local parts store
or
Less than $100 From A-Premium Alternator: Click Image for more details

Alternator Compatible with Ford Contour 1998-2000 & Mercury Cougar 1999-2002