First off, I didn't care about subtraction because Kirchoff's law is "summation". Secondly, in the passage below from the same book referenced above, you will see that for subtraction, you invert the vector and add it. Viola, there is the missing minus sign I have been talking about that lead you to perform subtraction. As I said way back at the beginning, you still got the correct answer, but you violated Kirchoff's law by subtracting, and the reason is because you redefined the voltage source instead of simply changing your point of reference.
Kirchoff's Law doesn't care if you sum your voltages clockwise or counterclockwise, but you can't do both at the same time, and that is what you tried to do in the original document in post #1 of this thread. Instead of simply saying that you changed your point of reference, you redefined the voltage source itself.
In your table in post #1, all of your voltages that you were summing were being reference counterclockwise until you got to Vcg, where you suddenly switched to clockwise. You hid this fact by putting in the equal signs between each operation. Had you not put in those equal signs, the mistake would have been much clearer to everyone that you changed directions from counterclockwise to clockwise.
Your phasor diagram makes no reference to the fact you changed how you define each phasor, and that is why it is wrong, and that is why the magnitude of the phasor between C and G needs a negative value to properly solve Kirchoff's Law.