In the diagram, I was trying to show the difference between a hydrogen bond and a covalent bond. If methoxide forms a hydrogen bond, it is weaker than a covalent bond. If it were stronger, it would be methanol.
I have gone through this before, but the arguments are different than in your textbook. Every pair of electrons is minus two. Hydroxide has three pairs of electrons. It can abstract a proton from an acid with pKa 14. Adding a proton does not change the charge of the non-bonded electrons. However, the remaining electrons cannot abstract a proton until the acid has a pKa -2.3.
Even though the net charge has changed, no change has occurred on the remaining non-bonded electrons or the nuclear charge of the oxygen. Remember, the non-bonded electrons of ammonia are more easily protonated than the electrons of a fluoride ion. It isn't net charge, it is the local environment of the electron pair. You can look at the bond lengths and determine the electrons extend further from the nucleus of a nitrogen than from fluorine. Also, even though ammonia has one more proton, the three protons are much further away compared to those in the nucleus of the fluoride ion. This is an inverse square relationship, 2x distance is 1/4 force.
If you are following my reasoning, then protonation must also shift the distance of the non-bonded electrons to account for the change in their basicity. Otherwise, all of the electrons of hydroxide could become protonated at a narrow(er) pH range. That doesn't happen. The bond lengths do change, but only by a small amount. But because we are comparing very small differences and the inverse square law, they have a big impact.
The net effect is that if we compare the non-bonded electrons of methoxide vs methanol, the non-bonded electrons extend further and are more basic. The process of converting a hydrogen bond to a covalent bond shifts all of the electrons. Methoxide has stronger hydrogen bonds because all of its electrons extend further from the nucleus. A very strong hydrogen bond is called a covalent bond.
The electrons of methoxide should be more reactive. If they react with water or another methanol, it converts methoxide to the less basic form, methanol. We can think of a hydrogen bond as an intermediate between a free methoxide and methanol. A reaction will occur most likely on electrons that are furthest from the oxygen nucleus (methoxide). Hydrogen bonding will compete with formation of the substitution transition state. They are equivalent in that sense. Hydrogen bonding will reduce the concentration of 'free' methoxide, hence a faster rate can be achieved in an aprotic solvent. Hydrogen bonding will be less strong for methanol because it's electrons are now closer to the nucleus and less reactive. Hydrogen bonding should only have an effect upon the rate by reducing the amount of free methoxide, but it will not alter the reactivity difference of methoxide v methanol.