There are four distinct cases of 未然形+う: intention, plus three of irrealis moods: hortative, dubitative, presumptive, . Very often Japanese grammar books refer to all of them as volitional form. Linguistic volition is defined semantically or syntactically, or combination of the two. So, 未然形+う is clearly morphological phenomena and therefore cannot be called volitional. If one must call it that way the actual term should be volitive (one of the irrealis moods), but it cannot be called such because it does not correspond to the usage of 未然形+う.
Volition, both in the vernacular and in the specifics of academic terminology, is about intent.
- The example above is about the speaker stating their intent to do something together with the listener.
- The dubitative example seems more of a suggesting hortative, as there isn't anything being doubted:
Shall we go?
... but if we extend that to a person talking to themselves, such as 「今日買い物に行こうかな」, that is still about intent.
- This does appear to be a distinct sense, separate from any question of intent.
He is probably a teacher.
Intentive / Intentional
- Separately from the examples in the comments above, there's also the straightforward intentional use case.
So we have three use cases about intent, and one about supposition or presumption. It is thus not surprising that English-language materials often call this form the volitional. See also the Wikipedia article about Japanese verbs, specifically the section that describes the volitional. This includes mention of the conjectural and hortative uses mentioned in the initial post.
Please comment if the above does not address your question.
I would say probably because such a piece of grammar doesn't exist in English in the exact same way it does in Japanese, therefore there is no one word to describe it perfectly. You could explain it more precisely, but that would in tern make it more complex. Normally something more complex wouldn't be taught until later on in the learning of the language, but, this grammar being so common and integral in basic speech patterns, has to be somewhat simplified just to help people get their foot in the door. The same way that the usage of だ/です is in reality much more complicated than it's normally explained; you can't jump right into something like that. And really, for all intents and purposes, it's fine without going so deep.
I think @rgolden has a very good point about simplifying a complex grammatical concept for beginners. If the authors of a Japanese grammar book are linguists (and I assume you are specifically referring to books where the intended audience speak English as a primary or secondary language), using technical terms to describe differences that many native English speakers cannot readily identify would not be very helpful. If the authors are not linguists, then it's possible they are just misinformed.
In the end, the result is the same: The learner is introduced to a common grammatical construction in a way that they can readily understand and apply. If the learner is interested in going deeper, they will eventually need to consult Japanese grammar books written for Japanese speakers.