The way I understand all this after reading through a lot of the comments here as well as some other places (I actually came across this question from a google search and found also a reddit thread on it and a steam thread) is that the naming of cRPG is not exactly correct in terms of Computer Role Playing Game anymore.
Again, this is from what I gather from things I read but also to what makes sense to me after all these years.
cRPG came to be as a term in the early days of computer rpgs to distinguish between pen and paper table top what have you, and the computer driven rpgs.
At which point it continued as a term wrongly for a long time in my opinion. For a time I actually remember distinctly the cRPG name not existing at all (around the years 1998 to 2005 I'd say?) and the term RPG being used alone.
It then came back into use for games such as Path of Exile, Torchlight if I am not mistaken among others like Divintiy .
At which point I think it makes more sense to come to the conclusion that it now means Classic Role Playing Game in reference to the first fixed view role playing games we experienced in the late 90's.
ARPG is also a more solid term that came around a while later than the birth of such games, since Diablo was always called a Hack n Slash RPG (was into both D1 and D2 a lot back in the day, aRPG was never a term I saw used). Hack n Slash RPG is indeed a clumsy name, and ARPG simply fits better for that genre.
Most studios simply name their game "RPG" and be done with it.
All that aside, what makes an RPG an RPG is not necessarily the story in my opinion, but the character freedom one has. This freedom needs to be of course backed up by a good story, otherwise it backfires on the character as well limiting that freedom.
Games such as Mass Effect for example. Widely accepted as an RPG, when for me it never really felt like one. It don't matter how interesting the story is and how many options it has or multiple endings or what not, it simply (for me that is) means that it has RPG elements in it that are also done really well.
An RPG to my understanding is about role playing, this is all a matter of creating those roles or shaping them and character creation, which in end is what makes the entire character stats, skills and progression within a class meaningful. Mass Effect on the other hand did not have either a great character customisation neither character creation model, and as an RPG it suffered a lot from this in my opinion.
Where do we draw the line in what is an RPG and what isn't? There I think. Otherwise Max Payne is an RPG because it has a detailed story as is The Force Unleashed because it has different endings based on your choices.
Sure its a fine line, but creating a great story with different paths and options in it is not in my opinion what defines an RPG and a title that has this alone and lacks in the character creation aspect lacks in the Role Playing aspect which then simply makes it not an RPG for me and I think this is what it all comes down to.
Even in DnD. The core was not about the story, it was about playing roles, the ruleset, classes, etc. If some lousy Dmaster came along with a lousy story with little options and deapth, then we could not blame it on the games design and say "this is not an RPG", instead we would simply say "dude, your story s*cks".
Now do it the other way around. If a really dedicated Dmaster comes along with an awesome deep long story, however tells you that you can only experience it as Conan the Barbarian because hey, thats' who the story is for, then the problem is not the story, its that it is simply not a role playing game at that moment. Even if they go as far as giving you said options of "ok you are Conan but you can chose if you are a shield and axe wielding warrior, a two handed warrior, an archer, or a thief". Still limited within that same experience just alters some variables as opposed to one being able to create their own character.
That is what the core element to an RPG is in my opinion, and why although games like Diablo back in the day and PoE today can be considered RPGs, even though their stories are completely linear and stretching that meaning of an RPG thin, they still hold to its core and biggest aspect… character creation.
Thus a game like Fallout 3 could be in my opinion still considered an RPG, and not a FPS. The combat and experience being a certain way or another and being the largest part of the game does not alter the fact that in the end, you are playing your own character you created even within a linear story. We can just say at that point, in the case that a story is too linear, that it is simply lacking.
End of the day, these names get swaped and used and re-used all the time. Developers usually go more for just using the term RPG and thats the end of it from their end. Genres and sub-genres can be defined from there on to differentiate the experience, and I don't think these need to stick to the same naming they had for example 15 years ago.
There is a huge use of late from even such sources as Eurogamer, PC Gamer, IGN, etc, of the term classic-rpg, and I think this re-emergence of the term has people confused since the original use of it was indeed computer rpg hence why cRPG gets a little lost in translation now.
RPG is what governs it all.
Sub genres from there on could be cRPG, aRPG and some even go as far as using the term fpRPG for first person RPG's. I think that is too much. cRPG is a term that simply points to the now dated but still wonderful in my opinion top-down fixed camera role playing games like those of old. aRPG could indicate that although a game for role playing, it is much more action oriented hence lacks in story options and not a proper rpg. And plain RPG refers to all other 3rd person or 1st person RPG's since 3rd and 1st person is the norm of gaming more or less in our time.
My vote for cRPG as a term and how it should be used in our time is classic-Role Playing Game.