Why is "a 100% increase" the same amount as "a two-fold increase"?

Yes, the correct usage is that 100% increase is the same as a two-fold increase. The reason is that when using percentages we are referring to the difference between the final amount and the initial amount as a fraction (or percent) of the original amount. So, if something gets multiplied by two, it experiences a positive increase equal to 100% of the original amount. The confusion arises because the word "increase" is used differently in each case. In the first case we mean the change between initial and final value; while in the second situation we interpret the change as a multiple of the original quantity.

People prefer to avoid the "%" increase for anything more than a few percent, due to confusion it creates: lots of readers fail to realize the distinction between "increase by" and "increase to", and even these who do, make a double take to spot which one was used, especially with values exceeding 100 by not much.

So, is increase of production by 120% better or worse than making it 180% of the previous output? How much is 3000% above norm? Is it 30 or 31 times the norm?

And when you start adding confusion of percent relating to which value they talk about, this becomes a total horror: The production first grew by 50%, then dropped by 50%. Oh, no, it did not return to original value. Currently it's at 75% of the original. Five increases by 10% each are totally not equivalent to increase by 50%.

You are correct in your usage, but it may be preferable to avoid percent if you can use plain fractions and multipliers instead. And on top of that, ALWAYS make sure you give the reference point and scale whenever not obvious, if using multiples and not direct values.

Process this: Today the weather is 15% colder than yesterday.