It took a bit of reading, but I think I see the source of our misunderstandings.
The article says, "Electrons in an #s# orbital have a greater shielding power than electrons in a #p# or #d# orbital of that same shell … For example, consider an atom of carbon whose electron configuration is #1s^2 2s^2 2p^2#. The two electrons in the #1s# orbital of C will do a better job of shielding the two electrons in the #2p#orbitals than they will of shielding the two electrons in the #2s# orbital."
The article is really saying, "Electrons in an #s# orbital have a greater shielding power than electrons in a #p# or #d# orbital of that same shell for electrons in the next higher shell".
It isn't saying that a #2s# electron is more effective at shielding another #2p# electron than a #2s# electron.
It is saying that a #1s# electron is more effective at shielding a #2p# than a #2s# electron.
This explains the dip in ionization energy on going from Be to B. When you go from a #2s# orbital in Be to a #2p# orbital in B, the #1s# electrons are more effective at shielding the #2p# electron of B.
