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. 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? So therefore, the force is dependent on the location of the of the object with respect to the fluid.
Please help me where I'm going wrong. I'm a beginner student.
The buoyancy force an an object is equal to the weight of the fluid displaced by the object.
The volume of displaced fluid is equal to the portion of the object's volume that is submerged (in, say, $m^3$).
The mass of the displaced fluid is the density of the fluid (in, say, $kg/m^3$) multiplied by the displaced volume. The remaining units are kilograms.
The weight of the displaced fluid, then, is the mass of the fluid multiplied by the acceleration due to gravity (in $m/s^2$).
The buoyancy force, then, is independent of the location of the object with respect to the fluid. It gets a little tricky if the density of the fluid changes with depth, as calculating the mass of the displaced fluid then requires an integral.
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.
This is not a full answer but Archimedes principle only works out if both the fluid and the object are in equilibrium. Even with that said, there are some exceptions, if an objects is stuck in a hole in the wall of a dam the force would not be vertical as there would be a component in the direction of the horizontal direction (as the fluid covers partially the object).
Your definition of Archimedes principle is a bit wrong but to answer your question:
Yes, the buoyant force is independent of the location. You will get the same amount of buoyant force depending on how much fluid you are displacing. But the pressure is different with height and due to other factors. Therefore, do consider that aspect when experimenting and observing.
