I am not sure how to ask this question but I am learning about
potential energy (high school physics) and from the deffinition of a
potential energy (energy stored in an object with the potential to
convert into other type of energy)
this definition of potential energy, sounds more like the energy itself. More below
i don't understand how for eg. when we have an object (let's say a
ball) on the ground, it has zero kinetic energy and also zero
potential energy and now let's say the ball starts falling of a cliff
so it will be gaining kinetic energy but when it had no potential
energy, how can it be now gaining energy?
indeed, in the planet+object system, the true 0 potential, would be better put at the center of gravity the earth if the reference frame is attached to the earth, if the reference frame is attached to the object, then it should be on the center of gravity of the object, and if the reference frame is neither, the 0 potential should be at the center of gravity of the object+earth system.
Also another question that was already discussed in SE (but i didn't
find my answer there) is why do we talk about the potential energy of
a system (ball+Earth) but kinetic energy of an object (ball)?
Because you need to define what type of potential energy we are talking about, in this case it's gravity (could be electrical for example)
We now see that potential energy is relative to a type of force (gravity), reference frame, and the final nail in the coffin of your definition of potential energy, is that it's not "stored in the object" . Potential energies depend on the spacial position of the object in the force field.
I hope all these elements will help you understand potential energy. Meanwhile your teacher probably gave you a simplified version of this that is enough to solve the problems you are supposed to solve for now. The main idea the teacher probably wants to convey is that energy is conserved, and that like voltage, we are more interested in differences in potential than absolute potentials.