Let's go back to our ball example and consider the following. You define the lowest point as y=0 and the highest as y=h. I define the highest point as y=0, so the lowest point is y=-h.

What do we agree on? What do we disagree on?

We agree on the value of the kinetic energy at any point.

We disagree on the value of the potential energy at any point.
We agree on the change in potential energy between two points.

This tells us that the change in potential energy is what really counts. That's what gives the change in kinetic energy.

We disagree on the value of the total energy (you say it's always equal to mgh; I say it's always equal to zero).
We agree on the fact that the sum of the kinetic and potential energies of the object is a constant value.

Now, this is something very important. In this situation the sum of the object's kinetic energy and potential energy is constant - that sum is conserved.

Let's define mechanical energy as the energy associated with an object's motion and its position. In other words, an object's mechanical energy is the sum of its kinetic energy and its potential energy.

Is this always true? Is an object's mechanical energy always constant?

We can think of lots of situations where this is not true, generally when other forces act on the object to change its energy, but let's write down a rule that applies when none of these other forces are present.

Principle of the Conservation of Mechanical Energy

Let's use the symbol U for potential energy.

As long as the only forces acting are conservative forces (forces that don't change mechanical energy) the total mechanical energy is constant.

ΔU + ΔK = 0

U₁ + K₁ = U₂ + K₂