How acceleration and gravitation (acceleration) are related/different?

Well, there's something called Gravitation acceleration which is the acceleration of an object due to a gravitational force. And maybe the difference is that gravity connects with the force of pulling downward, while acceleration can be increasing, possibly going up, forward, side to side, etc

The source of acceleration (without gravity) is transfer of energy from one object to another or change of its own energy structure so that one of its components changes into increasing energy of motion.

But gravity occurs with no measurable energy transfer nor is the gravitating mass changing its energy structure in order to produce gravity.

If the cause of acceleration and gravity is not related at all is the Equivalence principle just a coincidence that the resulting effects are physically identical and indistinguishable