What I want to understand, is what is voltage in terms of ball-like ions, forces and such intuitive entities.
First you'd need to clarify the situation in your drawing: where exactly does the voltage appear?
Answer: it's all in the microscopic gap between each metal surface and the electrolyte.
In other words, if you dip a chunk of copper into a bowl of salt solution, a constant voltage appears, with the metal charged negative and the liquid charged positive.
In other words, all batteries actually contain two batteries: one on the surface of one electrode, and the other on the other electrode surface. Google "half-cell" to find info on this.
Simplified intuitive explanation: when metal touches water, the polar water atoms start rapidly dissolving the metal. The water atoms surround metal atoms and yank them out of the surface. The metal starts dissolving rapidly, as rapidly as salt or sugar. But something else happens too, otherwise metals in water would vanish within minutes.
When water pulls a metal atom out of the metal surface, it ends up with a positive metal ion. Metal atoms have a 'loose' outer electron which becomes part of the bulk electron-sea of the metal and does not adhere to each atom. So, as metal atoms (positive metal ions) leave the surface and move into the water, the water acquires an enormous positive charge. The metal surface is full of left-behind electrons and develops an enormous negative charge.
In other words, when a blob of metal touches a blob of water, we end up with a self-charging capacitor!
Now obviously this capacitor cannot charge up to infinite voltage. The "dielectric" is a thin layer of water molecules separating the positively-charged water from the negatively charged metal. A few volts across a nanometers-wide dielectric will give us an e-field of megavolts/cm. This strong field pushes backwards on the positive metal ions and slows the corrosion process. Fewer ions leave the metal, the voltage rises more slowly, and finally we have equilibrium where a few ions are randomly falling back to the metal, and an equal number are still being dissolved.
So, a hunk of metal in a bowl of water doesn't dissolve in minutes, but instead develops a significant constant voltage where the metal is negative and the water is positive. (And, if you could somehow reduce this voltage, the corrosion process would run wild, and the metal would rot away as you watched. Or if you INCREASED the voltage instead, you could force any ions already in the water to plate back onto the surface, and the hunk of metal would start growing larger.)
What then is a battery? If you put two hunks of copper in a bowl of water, both hunks charge up to the same negative volts (a few volts wrt the water,) so if you touched them together, nothing would happen. Ah, but different types of metal will spontaneously charge up to different voltage. Touching them together then decreases the water-metal voltage at one surface, and increases it for the other. Try copper and zinc. WOW! They get hot, and one hunk starts dissolving furiously, while the other starts growing larger.
This would be very magical if noticed in centuries past, since the metal-eating reaction only happens when the two metal objects bump against each other ...or when both are touched against a 3rd piece of metal.