If you think of water pressure from a tank to a tap as a "push" then potential difference is also making such a push. Well yeah ambiguity from colloquial terms exist here, but it is not between push and pull. The two points making potential difference, "pushes" amps through the wire connected in between them. You can say that is not a push but like a chain being rotated. The volts is doing the rotating aka pushing+pulling of the chains. But saying a device pulling amps is essentially that devuce becoming less resistive to the push done by volts

If a device consuming power is pulling amps, then consider a device generating power is pushing amps.

You can push electrons into a capacitor if you don't connect the other side. A few of them, but more than zero.

There are electrostatic generators that work by pushing electrons on one side and pulling from the other of some insulator that moves between those two. But you can't sustain a steady state of only one of those actions.

You get it. Pushing AMPs is just a voltage significantly higher than the voltage required to jump a gap, or short a circuit, ect. Depending on your application. I like to think of it as a water slide, the source is the high end, the circuit is the slid, and the sink is the bottom of the slide and pushing AMPs is attaching a comic book villan's version of a pump big enough to drain all the water from the ocean.