I don't think that threefold division works that well, because 'engelang' is so radically under-defined, "'auxlang' operates on a different dimension (the dimension of practical use, which is unrelated to the features of the language - for any type of language, someone's suggested using it as an auxlang) and "artlang" basically just means "not either of the other two".
I think it would be more helpful to recognise two different levels of categorisation. On the one hand, you have what we might call a taxonomic classification, designed to be able to categorise all possible things; on the other, you have what we might call a generic classification, designed to highlight the most notable trends. For instance, saying whether something's a symphony or a symphonic poem is a generic classification - these terms highlight two common but distinct forms of music. But there are a lot of things that are neither, and quite a few things about which it's hard to say which of the two it might be. Whereas saying that something is chamber music or orchestral music is taxonomic - it describes almost everything without much dispute.
To give a linguistic analogy: a generic classification is like a sprachbund: it points out a cluster of like things, but doesn't pretend to include everything, and will often be non-exclusive. Whereas a family is a taxonomic classification: everything belongs to some or other family.
What the 'Gnoli triangle' is getting at is that there are certain vaguely recognisable genres of conlanging currently in vogue. The three most common genres, or 'schools' or 'traditions' are naturalism, idealism, and internationalism. But trying to classify conlangs taxonomically on this basis is like dividing music into "concertos", "classical Chinese court music" and "things played on the radio"...
Naturalism aims to create a viridical language that, as much as possible, could pass for 'real' - it emphasises probability (improbable features are permitted, and often encouraged, but only against a background of 'normality'), and the technique of diachronic derivation (in the naturalistic school, even a priori languages are expected to look as though they could have derived normally, and generally at least some level of diachronic depth is encouraged), as well as the idea of the language as something embedded in a cultural and socioeconomic context (questions like 'where is it spoken?', and 'who speaks it?' are common); specific generic tropes like the "linguistic report" (a description written as though by a linguistic researcher exploring a real language) aren't necessary but aren't uncommon, and in general the whole project tends to be shrouded in make-believe. Naturalism is the dominant genre of conlanging in the CBB, ZBB and many other online hobbyist venues, though it's historically novel.
Idealism aims to create a language closely inspired by concepts of the ideal language analysis movement. Design aims typically involve the reduction of ambiguities in vocabulary and in syntax, and the intentional avoidance of inessential tropes of real human languages. Methods are expected to be informed by at least some understanding of logic, mathematics and/or computer science. The aesthetic is generally brutalist. There is little concept of the language ever being spoken, and there is generally no supportive fiction; the language is presented in a purely objective fashion.
Internationalism is a school that aims to create a language that is proper to the human race as a whole, stripped of its cultural specificities. Internationalism evolved out of the auxlang movement, but in reality most internationalist conlangs are not created with any real intent at practical adoption. They emphasise universalism - common techniques involve pluralistic borrowing (often formalised and probabilistic) and levelling out (removing 'difficult' features that are less universal, or even trying to create vocabulary items that represent the 'common denominator' of two or more languages). There is often, though not always, a broader ideological element, with the language privileging universalist and egalitarian behaviours and disfavouring others. If there is any academic influence, it's from fields like sociology and statistics (rather than historical linguistics (naturalism) or logic (idealism)).
Taxonomically, you can create classifications based on any question you want. Here's two suggestions...
First, a taxonomy that tries to capture the trends represented by the above genres. For this, I'd begin with a binary distinction: mimetic, vs speculative. A mimetic language is one that attempts to mimic some real language or group of languages (including the group of 'all real languages). Mimetic languages, at least at first glance, attempt to look like real languages.
But not all mimetic languages are naturalistic. There's a tendency to see all non-naturalistic mimetic languages as just "bad naturalistic languages", but the intent is often different. Bogolangs fall into this category, for example - they're not naturalistically plausible, but they are in their own way designed to look superficially real. Similarly, Tolkien's languages. They're not part of the naturalistic school, quite explicitly - the Eldar were not bound by human linguistic laws, and, particularly early in their proposed diachronics, things happen in them for no reason other than that the loremasters felt something seemed cool. Sindarin is also perhaps too coarsely inspired by Welsh to be really naturalistic. But these are mimetic languages. And then there's Klingon - informed by linguistics, sure, and superficially plausible, but too driven by extra-linguistic stylistic considerations and too filled with jokes to be a respectable example of pure Naturalism.
So, I'd divide mimetic languages - if I really wanted a taxonomy - into two types: those that prioritise directly mimicing human languages ("viridical"), and those that, while attempting to look superficially realistic, are driven more by the desire to create a specific impression on the observer or by other extra-linguistic considerations (a well-crafted joke languages, for instance) - these we might call 'stylised'. Quenya and Klingon are stylised mimetic languages. So are most bogolangs (though some, where the emphasis is less on a clear aesthetic and more on linguistic realism, are non-Naturalist viridical languages).
[Naturalist languages are all viridical. However, not all viridical languages adhere to the specific generic interests of Naturalism, like diachronic justifications, fictive speakers, or 'probability'. Esperanto is a famous example of a (mostly) viridical but non-Naturalist language. So are most Internationalist languages]
Meanwhile, speculative languages are those that do NOT attempt to mimic real human language even superficially, but instead have some other organising principle, which generally can be expressed as a what-if speculation. They're hard to subcategorise because they're so vast in scope. Idealist languages are speculative ('what if a language were perfectly 'logical'?'). So are languages to optimise comunication with computers ('what if a full language were like a computer language?'). So are languages for aliens, if they're treated as truly alien. So are pure jokelangs. So are most languages with a strong ideological element - Toki Pona, for instance, while having some Internationalist trappings, is probably better described as speculative than as viridical.
An alternative taxonomy: how about asking "who is this to be spoken by?"
If the answer is "real people", we're dealing with a practical language. Practical languages can be divided into futurist languages (languages the creator believes ought to be spoken by particular people in the relatively remote future) and reformist languages (languages the creator believes ought to be spoken by a group of people in the relatively near future). Futurist languages can be divided into task-oriented languages (primarily languages for computers or for computer-human interface) and general futurist languages (intended for a whole society in a particular condition, rather than for a specific task in the future).
Reformist languages are probably most usefully divided into universal and particular reformist languages - for everyone, or just for a particular group. Universal languages include Internationalist languages, but also many Idealist languages. More generally we could divide them into languages that should be spoken tomorrow by everyone because they're better - perfectionist languages (Idealists, but also, for example, radically egalitarian languages) - and languages where the proposed 'reform' is driven by other motivations (like the desire for a single world language) and the language is not inherently better as a language structurally, but for some other reason would be a good language to adopt - ease of use, neutrality, etc - which we might call pragmatic languages.
Particular languages, meanwhile, have some motivation specific to a particular group. We could divide these into unifying languages (which are to be spoken by a group that doesn't currently have a common language) and distinguishing languages (to be spoken by all or some of the speakers of a particular current language). Unifying languages include things like lingua francas, jargons and creoles (caused by combining elements explicitly), and panethnic reconstructions (where related languages are in some way reduced to a proposed new common form akin to all but biased toward none). Distinguishing languages include things like egalitarian reforms, purification movements (stripping out 'foreign' influences), resurrections of dead languages, and secret languages (adopted to create a new in- and out-group).
If the answer, though, is "fictional people", we're dealing with a fictional language. The next question is probably 'are they human (or human-like)'? So we have human-like and non-human-like languages. Human(-like) languages can then be divided into languages for groups that don't exist, but relatively easily could have done (alt-history languages), languages for people who could theoretically have existed on some version of Earth (and may have done!) but there's so little known that they could have spoken anything (a Neanderthal language, or an Atlantean language), and languages intended for people who live in another world or on another planet.
And the third answer is "no-one", in which case we're dealing with a theoretical language.