Based on that description, I would also think the infinitive is morphologically required by the surrounding context. But I'll try to make an argument for the opposite view.
In syntax, I've heard these called interpretable features (markings that are semantically part of that same word and used when interpreting it) and uninterpretable features (markings that are attached by agreement, in order to match the interpretable features of something else in the sentence). This is slightly broader than Booij's morphological definition, because I would say that e.g. German nouns have an interpretable gender feature, even though it's usually not explicitly marked on the noun. But it is something that you choose for the noun (the speaker can convey different meaning by saying See (m) "lake" vs See (f) "ocean") that then applies uninterpretable markings to other parts of the sentence (der See, die See).
In this model, I would call most English verb inflections uninterpretable on the verb, specifically because English verb inflections seem to come from some other head than the verb itself. That's why the T (or I) category is posited in English syntax:
I Tpres like this.
I Tpast liked this.
I do like this.
I did like this.
In other words, we posit this special category because English tense marking can be "pulled" out of the verb phrase, if something else appears earlier to receive it. This is hard to explain if tense is a feature of the verb itself: how could it be torn away like that?
This tense element also seems to be outside the verb phrase itself, because it stays in its place when we move or elide the verb phrase:
I said I [finished the job], and [finish it] I did.
*I said I [finished the job], and [finished it] I.
I voted for Bob, and she [voted for Bob] too.
I voted for Bob, and she did [] too.
*I voted for Bob, and she [] too.
Thus, English syntax is often analyzed as having a "T" (or "I") element that's not always visible, which can either be visible like "did" (and take the inflection itself), or invisible like "Tpast" (and pass the inflection off to the verb).
And it's this element that the feature of tense belongs to, semantically; the verb is just where it's shown on the surface, like how gender semantically belongs to the German noun but is pronounced on the article instead. Of course, morphologically, you could consider the T and the verb to form a single unit together, in which case the unit "liked" (consisting of Tpast and "like") has an interpretable past-tense feature on it. I imagine this is how Booij would analyze it.
So what about the infinitive "to"? Well, it seems to belong to this same "T" or "I" category, since you can move or elide its verb phrase and leave it behind. And it enforces a specific inflection on the verb, rather than inheriting a form from the matrix clause:
She told him that she looked at it.
*She told him that she looks at it.
*She told him to looked at it.
She told him to look at it.
So "to", or "to look" if we're taking T+verb as a unit, clearly has an inherent tense feature on it, from a syntactic point of view.
That's why I would call the infinitive marking an inherent inflection, based on my understanding of Booij's system.
