From what I got out of Archimedes principle, it essentially states that the buoyancy force is the weight of the fluid in which the object is submerged in.
This is not quite right. Archimedes principle states that the bouyancy force is the weight of the fluid that was displaced by the object.
But in my general physics class, it states that the buoyancy force equals the density of the fluid times gravity times the volume in which the object displaces?
This second statement is correct because density times displaced volume is displaced mass. The displaced mass times gravitational acceleration is the weight of the displaced fluid.
In the special case where the object is fully submerged, then the volume of the displaced fluid is the volume of the object. In this case, it doesn't matter where the object is, just as long as the object is fully submerged, the volume of displaced fluid is fixed (= the volume of the object).
However, when you have an object that is partially submerged, then the location does matter, because the deeper the object dips into the fluid, the more fluid is displaced.