In isolation, the definition of the p-value indeed appears confusing and unmotivated. However, when viewed in the context of hypothesis testing, it's use and intuitive motivation become, hopefully, more clear:
A p-value is nothing more than a standardized test statistic. It is used in exactly the same way that the Z statistic, T statistic, Chi-squared statistic, and myriad other test statistics are used, and for the same purpose. The only difference is that p-values can be directly compared to a desired Type I error probability ($\alpha)$ when determining whether to reject, whereas the other statistics require distribution-specific values indicating when you can reject at a particular $\alpha$. That is all there is to a p-value.
The reason the p-value is a standard or universal test statistic stems from the fact that you can use the CDF of a sample statistic to specify the rejection region in one of two ways: Either specify the values of the test statistic itself that capture (1-$\alpha)$ of the probability (i.e., a range on the abscissa of the CDF), OR you can "invert" the CDF to get the equivalent range on the ordinate of the CDF; this inverted distribution is the distribution of the p-value for a given value of the underlying test statistic. It is just as valid as the actual sampling distribution and, because a CDF is a function, there is a one to one correspondence between a given range of the p-value statistic and the range of the underlying test statistic, so the two specifications of a hypothesis test are equivalent. Again, the only difference being that a p-value has units of probability, and so its value has a universal interpretation, whereas, for example, a chi-squared statistic value cannot be directly compared to $\alpha$, it must first be converted to a p-value or a pre-calculated rejection value specific to the Chi-squared distribution - a value that cannot be used for other tests.
I hope that clarifies the proper use and interpretation of the p-value. You are correct that the p-value statistic does not give you the type I error probability for this test, since according to Neyman-Pearson theory, a Type I error is associated with a test methodology not a particular test. IF you specified a rejection region as "Reject when the p-value is $\leq 0.05$" then you would be correctly using the p-value to specify a Type I error, since you are just re-stating the rejection region of the underlying test statistic in terms of p-values. However, post-hoc use of the p-value, just as with other questionable "after the fact" analyses ("Retrospective (observed) power" being an infamous example), invalidate the p-value as a measure of Type I error, since you only decide on the rejection region after you took the data.
So, the bottom line is to think of the p-value as just another way of specifying the rejection region, one that uses probabilities instead of test statistic values. That should ensure that you don't try to use it in inappropriate places, which can happen if one focuses too much on the p-value as a measure of "rareness or evidence" vs. simply another test statistic that requires a pre-determined rejection value.
