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There are four distinct cases of 未然形+う:
[intention](https://en.wikipedia.org/wiki/Intention), plus three of [irrealis
moods](https://en.wikipedia.org/wiki/Irrealis_mood):
[hortative](https://en.wikipedia.org/wiki/Hortative),
[dubitative](https://en.wikipedia.org/wiki/Dubitative_mood),
[presumptive](https://en.wikipedia.org/wiki/Presumptive_mood), . Very often
Japanese grammar books refer to all of them as **volitional form**. Linguistic
[volition](https://en.wikipedia.org/wiki/Volition_\(linguistics\)) 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](http://wikidiff.com/volition/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.
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terminology
Why do English speaking learners of Japanese use term "volitional" to refer to 未然形+う?
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.
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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.
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**Volition** , both in the vernacular and in the specifics of academic
terminology, is about **intent**.

Hortative

  * The example above is about the speaker stating their intent to do something together with the listener. 

> Let's go!  
>  行きましょう！

Dubitative

  * The dubitative example seems more of a suggesting hortative, as there isn't anything being doubted: 

> Shall we go?  
>  行きましょうか。

... 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.

Presumptive

  * This does appear to be a distinct sense, separate from any question of intent. 

> He is probably a teacher.  
>  多分先生でしょう。

Intentive / Intentional

  * Separately from the examples in the comments above, there's also the straightforward intentional use case. 

> 今晩テレビ見よう [と思う]。

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](https://en.wikipedia.org/wiki/Japanese_verb_conjugation#Volitional,_presumptive,_or_hortative).
This includes mention of the conjectural and hortative uses mentioned in the
initial post.

* * *

Please comment if the above does not address your question.
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