At the top of exercise 7 on modulations, there is this text: augmented note: VII. degree - diminished note: IV. degree.
I am not sure I understood what it means: that if I find an augmented note, it is to be understood as the VII degree of the new key, and if I find a diminished note, it is to be understood as the IV degree of the new key?
It seems to make sense to me: an F# in C indicates that I have modulated to G (of which F# is the seventh degree), a Bb in C indicates that I have modulated to F (of which Bb is the fourth degree). From the altered note, one must then step back to find a V -> I (D-G, or C-F) or a V6 -> I (F#-G, or E-F) to establish where the modulation begins.
Personally, I would advise you to think from a more practical point of view: with my students, I use Franco Margola's "Practical method for bass harmonization without numbers," which is the same method I used at the beginning of my composition studies. It is an inexpensive, practical, progressive little book...
Coming to the problem you raised, the issue can be resolved like this: every modulation (or tonicization, as it would be better to call transient modulations) is characterized by one or more signal notes, i.e., those notes that indicate a modulation is about to occur. Signal notes can be explicit (meaning they are found in the given bass line) or implicit, meaning they are inserted into one of the 3 upper voices. Signal notes can be raised scale degrees or lowered scale degrees.
If, for example, we are in C major, the possible raised notes (which we could call transient leading tones) are: C# (to modulate to the second degree, D minor), D# to modulate to the third degree (E minor), and F# to modulate to the fifth degree. On the other hand, the degree that can be lowered to modulate is often the seventh; if we are in C, B becomes Bb to modulate either to F major or to D minor. The lowered sixth degree, however, has a more complex treatment because the concept of the augmented sixth is also introduced.
So your reasoning is not wrong; I would simply invite you to be as practical as possible—think about harmonization first, and then the terminology will follow. That is, you must think about the signal notes in the case of modulation; they are what help you better understand the concept of modulation.
