Why 10Log(p1/p2) and 20Log(v1/v2)? Why not just 10Log(x1/x2)?
In other words, in my head, a doubling of power or doubling of voltage
  is the exact same thing!


Incorrect - doubling the voltage quadruples the power. Moving on....

Decibels are all about power ratios.

In your question you say you want to calculate the voltage (or current) db ratio. This is where you are getting confused.

For instance if I said the "signal" across a resistor increased by 6.02 dB, the power will have gone up four times and the voltage will have doubled.

On the other hand, if I said the voltage had doubled, in dB terms this is: -

20log(2) = 6.0205 dB and if I said the power had quadrupled, in dB terms this is: -

10log(4) = 6.0205 dB. Does this make sense?

The basic problem is that you believe that there is such a thing as "voltage (or current) db ratio" and that somehow this is independent or different to power.