I suppose this is one of those issues that certain people have strong feelings about, and yes, there have been corporate lawsuits over exactly who can call what an M4, etc.
The common sense rule of thumb in these parts seems to be that terms like M4 are describing a general configuration, and it isn’t terribly important whether the weapon was made by Colt or a competitor – some of the latter being far better than others, of course.
In my mind, an “AR-15” refers generally to a 20" barrelled semi-automatic rifle with a fixed buttstock.
16" barrelled carbines with telescoping buttstocks are probably best described as “AR-15 Carbines,” as Colt set this common-sense precedent in their own marketing.
Others may disagree, of course, but when I hear “M4,” it calls to mind one of two things: in select-fire configuration, a 14.5" barrelled carbine with the characteristic stepped-barrel for installation of the M-203, or in a semi-automatic configuration, a 16" barrelled carbine with the same barrel profile.
Shorter barrels (NFA guns) are generally referred to by the nomenclatures that Colt established for weapons of this class (i.e. “Commando.”), or simply “SBRs.”
So, that begs the question: is a LMT or CMT carbine with a stepped-barrel an “M4?” Technically speaking, no – and that’s where terms like “M4gery” are sometimes used – but in practical terms, the term M4 is accurate enough as a general description. If someone flames you for that kind of basic usage, then they are probably just being elitist and/or unnecessarily disagreeable. It’s not a big deal for most of us, so long as the term used correlates to something that we understand on more-or-less baseline terms.
If you want to get really technical, you could always differentiate between A1, A2, A3 and A4 specification weapons with such qualifiers as “Government Carbine” and such, but again, these terms follow Colt’s lead. Other vendors will name their models with a wide variety of alternative designations, of course, and most of these are not especially useful unless the masses are likely to have some familiarity with the actual weapon being described (a good example of the latter would be the HK 416, or to a lesser extent, the S&W M&P-15T or Noveske N4).
In any event, this is why it has become common practice to refer to a Bushmaster carbine with a stepped-barrel as an “M4” in loose terms, even though it is not – and never will be – a true M4. If that isn’t confusing enough for you, consider that Colt doesn’t even refer to their semi-auto M4 as an “M4,” but instead calls it the “Law Enforcement Carbine” … which is why many owners will specifically state that they are talking about a “LE6920” to eliminate any confusion. I’m really out there on the fringes with a LE6920HB. 
Does any of this really make a difference? Not really, but with so many variants, configurations and manufacturers out there in the AR business these days, we’ve had to adopt some unofficial de facto standards to communicate effectively, and as stated, the fallback position in all of this has usually been to go with whatever Colt named a given model. (I won’t go into the fact that Hartford has long made it their practice to vary things quite a bit even within their own naming conventions.) The overriding point to all of this is simply that the “M4” designation has gone mainstream to the point where it is really a reference an entire class of similar products, much like what has happened with the terms “Kleenex” or “Xerox” over the years.
So, to paraphrase Danger Close, if you’ve got a carbine that looks like an M4, shoots like an M4 and causes you to spend absurd amounts of money on ammo and accessories like an M4, no one here is likely to split hairs with you if it is has something other than a rampant colt on the receiver. The term “M4-type carbine” may be more useful, but it really depends upon who you’re talking to, and what the context of your discussion might be.
Chief