[Coda]

Technically it IS ΔV = IR, because it's measuring the voltage drop across a resistive load, but when you're working with electric current you never have to deal with absolute voltage, so nobody bothers writing the delta.

The equivalence of the uniform field equations to Ohm's Law implies that having a circuit connected across a potential difference (such as the terminals of a battery) creates an electric field constrained to operate mostly inside the wire. (It's not perfectly contained within the wire, just like a field between two plates still has a field OUTSIDE the plates that isn't uniform; this turns out to be very important because that's how inductance works, which is how we can build transformers that can trade current for voltage while keeping power constant.) But BECAUSE it's constrained like that, it's really not worth TREATING it as a field for the most part. One of the biggest reasons it's not worth it is because distance really doesn't MATTER and resistance is the relevant proxy for that.

And the electrons are moving because they're present in the circuit and there's a potential acting on them, so in that sense it's not different.

Remember, it's both pushing (negative repels electrons) and pulling (positive attracts electrons), and that's why the force is uniform throughout the field / circuit.