Or it's the logarithmic function lg(x), where the independent variable is the score and the dependent variable is roughly proportional to the number of times you've clicked.
"Therefore, despite these two functions being the inversely related, the correct relationship is 2^x."
There is no "correct relationship", there is a relationship that (depending on choice of variables) we can describe as y=2^x or y=lg(x).
However, as you said, one choice of describing the relationship is certainly more natural. The exponential description assigns a score to every natural number. While the logarithmic description does not.
I would interpret correct in the way that it is sometimes said, "the correct way to think about X is..." which doesn't say other ways of thinking are wrong, but limited or not illuminating.