It has been many years since I have used this information (so please forgive my inaccuracies, but I wanted to get you PART of the way there with an answer (since no one has responded yet). The probability density is the integral (area under the curve) of the probability. I think the reason for the funky discrepancy between the two definitions is because..
Let's forget about wave functions for a second, and just use $x$. If we were to integrate $x$, we would get $\dfrac{x^2}{2}$.. which can be rewritten as $$(.5) * x * x$$
the $.5$ can be seen as a simple scaling factor, and therefore the integral is PROPORTIONAL to
$$x * x$$
Now, if we substitute a wave function (or probability function) in for $x$, we should get a similar result.
