Should be P = V^2 / R, but other than that, yes exactly.

Everyone saying "it scales with voltage" or "it scales with resistance" are describing different sides of the same coin. At the end of the day, the thing we're talking about is just power, you can pick and choose whichever version of the power formula that you want.

Yes, but the power dissipated by a wire is, roughly speaking, a function of current, with the voltage across the wire depending on the current and the wire resistance (Ohm's law). For a wire, you use P = I^2^ R, and for a load, you use P = V^2^ / R.

In other words, the current through a wire is dependent only on the load and its power supply, while the voltage across it depends on both the current and the size of the wire. Hence, it's cleaner to talk about the current

This is why AWG charts always give a current rating, not a power rating:

Joule's first law (also just Joule's law), states that the power of heating generated by an electrical conductor equals the product of its resistance and the square of the current. Heat is a function of current, not total power.