$p$-Values: Standardized coefficients vs. partial $R^2$ vs. $\Delta R^2$

In this excellent answer, the definitions of, and differences between, the three quantities in the title of this question are laid out.

My question concerns the relationship between their $$p$$-values. This answer states that the $$p$$-values of a standardized $$\beta$$ and the corresponding partial $$R^2$$ is the same. My question is two-fold.

1. Why is this true?
2. If it is true, what is the relationship between this $$p$$-value and that of the corresponding semi-partial correlation coefficient?

In response to the comment from @ttnphns below, I ran an example on the duncan_prestige dataset. One does in fact see that the $$p$$-values for standardized $$\beta$$'s are the same as those for the partial correlation coefficient so question 1 above has been clarified.
But notice now that the $$p$$-values for the semi-partial correlation coefficient are in fact significantly larger than that of the partial correlation. (I was able to reproduce this behaviour in other datasets as well.) Why does this happen?
My intuition concurs with what @ttnphns claims below, but consider that the ppcor documentation (on which the python package that I used for my computation is based) lists the exact same formula (2.8) for the the $$t$$-statistic of the partial and semi-partial correlation coefficients; therefore, since these coefficients are de facto different due to their differing scalings, they will have different $$t$$-statistics and $$p$$-values (since $$\text{df}$$ is the same in both cases). Is this an error in ppcor or is something else going on?