I think your trouble stems from definitions. I don't think your definition of voltage is correct.
This is close but not quite right
That's it's units allright
but it's actually a simpler concept than that.
And Rive is right, when we lazy engineers get accustomed to using it we broad brush past the details.
Here's how to think about it
Voltage is Potential Difference. Two words, not one. Difference is the easy one.
So what's potential?
Potential is the work required to bring a unit positive charge from infinity to wherever you're measuring potential.
That takes some thought.
Imagine yourself at Alpha Centauri(close enough to infinity for demonstration purposes)
with a one Coulomb sized bucket full of charge,
a force gage, and a ruler.
I grew up with dynes and centimeters and ergs but Newtons and meters and Joules are easier...
Now start walking toward earth, measuring the force in Newtons exerted on your bucket of charges and tabulating it at every meter along the way.
So as you move toward Earth you're going to tabulate the Newton-meters and keep a running sum.
When you've reached the top of your lightbulb you will write there what is that point's potential. That'd be its absolute potential.
Now repeat but this time walk to the bottom of your light bulb and again write its absolute potential.
The difference between those absolute potentials is the voltage across your light bulb.
When i grasped the concept was the day I imagined myself counting dyne-centimeters all the way from Alpha Centauri to my workbench in Miami Central High School's electronics lab, ca 1962 .
Now since we can't get to Alpha Centauri let alone infinity it's completely impractical to do that measurement,
and that's why we never know what is the absolute potential of anyplace.
So we just have to settle for the difference in absolute potentials between two places we can reach.
That's easily measured with a two wire voltmeter provided its leads are long enough to reach our two points of interest.
That difference in absolute potentials is "VOLTAGE" . The voltmeter reads that.
Whatever is the absolute potential at one end of a battery, it's different at the other end by whatever is the voltage of your battery. We can only measure that difference.
That's voltage. Forget about clouds of electrons.
Now, an electric field will cause charges to migrate along the field if they can. Inside a copper wire they migrate easily so a miniscule field will cause quite a bit of current . That's why the voltage between ends of a wire is miniscule, charges move equalizing local charge densities along its length..
This oversimplified layman's explanation should help you make sense of the concept. I don't mean to come across anti-academic; au contraire.
Don't just memorize formulas, understand what's happening and they'll become intuitive.
Looking up definitions is always a good idea - laying the foundation if you will. Then use your imagination to link them to your everyday experience. That's called "Memory Pegs" .
Working inside circuits is different from electrostatics, we have simplifications like no field along a wire and V=IR neglecting magnetic induction.
Poynting Vectors and Magnetic Vector Potential come later, after you've got used to working inside circuits.
Good luck in your studies.
Apologies for being so basic in an academic forum, corrections to any of above are welcome.
old jim himself