The ordering changes due to interactions between the 2 s and 2 p orbitals. For O, F, and Ne, the 2 s and 2 p orbitals are quite far apart in energy and the lack of interaction gives the ordering as predicted by Figure 9.36. So they all agree in a very complementary way. One possibly confusing difference is that for Hybridization we created hybrid orbitals sort of as we needed them. Multiple bonds are always formed from one sigma bond plus one or two pi bonds.
Bond order is just another name for how many electrons are involved in forming a bond.