Every conlang is in the broadest sense an engelang (there will always be an engelang?) in that each is created for some extrinsic reason, even if that reason is as simple as self-pleasure (let's assume the pleasure is aesthetic not abusive). So we will need a more restrictive definition of the term 'engelang,' but hopefully not too restrictive.
The definitions of terms are inevitably going to be non-exclusive, since various factors achieve prominence in various definitions but all may play roles in the development of a conlang itself.
Given LCCIII, let's start with artlangs, for which the dominating reasons are aesthetic: the language is to be beautiful (or, I suppose, ugly) in itself in some way or to be a part of an aesthetic object (either prominently or poohbahly: “merely corroborative details to add a note of verisimilitude to an otherwise bald and uninteresting narrative”).
At the extreme of this are modlangs which consider only the language itself: phonology, morphology, syntax (and maybe semantics – but that tends to move to other considerations). This tends to be a very idiosyncratic field, creating the linguistically, perfect or beautiful ( or perfectly ugly or defective) language. Jeff Burke's remark about people being unable to appreciate his language because they did not know enough linguistics, though not really intended in this way, would serve as a slogan for modlangers. I don't think we can find pure cases of this sort easily, but I am sure that there are some: Tolkien's papers show signs of this most secret form of the secret vice.
At another extreme are texlangs, where all that matters is the appearance of language: the shape of text, the orthography and calligraphy, the sounds (rarer?). Of this we had many samples at LCCIII, in most of which the underlying language either did not exist at all or was irrelevant: the unreadable books, for example. I am inclined to think that glossolalia of the Azusa Street sort would fit here – if there was any notion that it was constructed consciously or even unconsciously; literal mumbo-jumbo clearly does belong.
But the most common artlangs are those where the language plays a role in some other art work: a book, a conworld, a game (we're not being exclusive even here, obviously). Of course, this overlaps texlangs somewhat, but let us distinguish these out by claiming that the specimens of this sort are meant to be meaningful or at least referential.
The minimal use of conlangs in other art forms is to provide names and other labels, nomlangs.. What we know of Liliputian and other languages Gulliver encountered is mainly what Lem was called. Similarly, in the works of Borges and Lovecraft we have names (well, referentials anyhow), including sometimes names of languages, but little else. What we have is usually nothing more than an implied phonology (really an unspecified adaptation of Latin orthography), usually designed for effect (harsh, eerie, alien certainly).
After that, we seem to be on a slippery slope for the differences are mainly a matter of how much the language is used in the work, from occasional bits of dialog (one sentence of Panlan in The Troika Incident) to the whole work being in that language (Clockwork Orange, Watership Downs, Ridley Walker). But it seems wrong to group together something like Tyrusian, with a couple dozen sentences (all that is needed for a miniseries and a couple of books) and no explicit apparatus, with some of Tolkien languages which have a good deal of text as well as worked out phonology, vocabulary, syntax and even history. Perhaps there is a something that could be done in terms of the attention given to the language: Tolkien devoted much time to his (indeed, occasionally claimed the books were just an excuse to use the languages) while Fischmann whipped his off in a matter of hours, expanding it as needed for the show.
Call these Litlangs and maybe fraglangs, though where to draw the line is considerably less clear than the extreme cases.
Auxlangs have a common goal: to be easy to use for international communication in some areas. Almost every language with more than two speakers gets suggested as such but many are not designed for this and not well equipped either (limited vocabulary without clear means of expansion, inherent difficulties for some cultures, etc.)
Philangs: everything has its correct name: phonosemantic congruity, from Cratylus on. Often these can only be written, as the phonosemantics makes for unpronounceable clusters.
Every one is, of course, automatically an auxlang, since the whole point is to get everyone (that counts) should use it to be clear and correct.
Reconstructed languages – the core of comparative linguistics –are scientific hypotheses, and so can be right or wrong. They are a posteriori by definition. And their creators would resent their being called conlangs, though many of the skills used to create them are similar to those for creating a conlang. (And, of course, a set of conlangs devised for the purpose, would make a nice controlled tool for a class in comparative linguistics –even with all the traps built in).
Revived languages: Hebrew, Manx, Greek, etc. (Biblical Hebrew and Sanskrit also) would also not be happy being called conlangs. But again, given the the a posteriori nature and the fact that they are usually the result of the work of a committee – but with one dominating figure usually, the skills are much the same.
Languages to test some linguistic “universal” or hypothesis. If the universal says two features can't go together, construct a language that has them both and see if anyone can learn it. If the hypothesis is that the realization of some feature of languages affects the way we think, build a language with a novel realization of that feature and look for novel thought patterns.
Loglangs use the syntax of, or otherwise, incorporate the structure of a logic, extended first order predicate logic, say. Or they may in some other way be designed to aid logical thinking
11. The distinction between a priori and a posteriori can occur anywhere (the unreadable book uses real radicals and other features of the Chinese writing system), a philang might take words or common sounds from natural languages (lots of languages use n in words related to noses so n- sound must be part of representation for nose, for example). The distinction seems to be mainly about vocabulary but can be for other features as well.