Well, Dolby specifically chose to use the term "Low Frequency Effects" instead of "Subwoofer" for the channel we usually associate with subwoofers for a reason. One reason was that some HiFi systems were already taking advantage of add-on subwoofers, usually in the form of a large dual voice coil woofer in a very large enclosure and a passive crossover. I was one of those going that route with a BSR woofer at first, and later a pair of higher quality DIY subwoofers which I also installed a very robust L-Pad on for the high pass output of the passive crossover allowing my to adjust the level of the [typically] high sensitivity full-range speakers. It was also specifically to ensure a speaker was present which could handle the very demanding low frequency energy directors wanted to use in their soundtracks to add impact to their movies when projected in Cinemas. So, the LFE channel was given to us, and we started using the term "Subwoofer" synonymously with "LFE Woofer".
Since there is no officially approved definition of "subwoofer", I tend to use the logic of what makes up most speaker systems since the advent of crossovers and the semantics of the English language. The most basic speaker is a single full-range speaker. A two way system is almost always consisting of a "Tweeter" and a "Woofer" (but sometimes the "woofer" is referred to as a "Mid-Woofer" if it is expected to operate well into the midrange when paired with a smaller tweeter). A three-way system usually has a "Tweeter," "Midrange," and "Woofer." And, when more speakers are added we hear terms like "Super-Tweeter," "Mid-Woofer," "Low-Woofer," and "Sub-Woofer." (I chose to leave off the term "Squawker" which was used for the larger midrange horn in the early days of this hobby.)
Since we have added the concept of an add-on low-bass speaker with its own amplifier and crossover which we are obliged to integrate into the system, we have always referred to that type of speaker as a "subwoofer," even when it doesn't get that low at all. I mean, go to Best Buy or dig up memories of shopping in Circuit City and tell me more than a quarter of the "subwoofers" for sale actually produced bass below 40Hz and tell me the term "subwoofer" in common parlance requires the object bearing the title actually produce sound below 40Hz!
I would argue that a "subwoofer" is any woofer with it's own amplifier and processing (like a crossover, EQ, or DSP), which is combined by the user with any large surround system or primary speaker(s). I also argue that it should not generally operate in the bass range where most listeners can easily perceive pitch and where the frequencies are necessary for us to recognize any sound. I also argue that a "Subwoofer" operate in a range which allows for creative placement without hindering stereo imaging.
Physical design
Since a main speaker system is typically self-integrated, meaning you buy a speaker system in which every driver is connected to a crossover (or internal amplifier) and the end user only has to place it in the room and hook up to an amplifier or preamp, I typically consider any bass-specific speaker which the user must integrate separately a "subwoofer." Since it is separate, is designed for calibration in order to ensure natural integration with nearly any main speaker system, a subwoofer can be from almost any company or of nearly any design. This, to me, is one clear differentiator of a "subwoofer" from a "woofer" or "bass-bin".
Musical Pitch and Recognition of Real Sounds
Using decades (even a century) of repeated and proven research we know that at a minimum humans generally need to hear a real world sound reproduced down to at least 200Hz to clearly recognize what we are hearing, and it that perception is significantly improved when the reproduced sound reached down to about 120Hz. Beyond the 120Hz general limit (there is variation in all listeners, especially between males and females), the speed and accuracy of perception is not improved, so I could argue that a subwoofer should not be expected to operate above about 120Hz.
Ability to Locate Speaker
The third aspect is how we integrate the subwoofer. Decades of research by Dolby, THX, Canada's NRC, Harman, and a swarm of academics and corporations, has demonstrated that in nearly all domestic (family home) indoor situations, listeners cannot pinpoint the location of a speaker producing sound below approximately 80Hz (which was the primary reason Dolby chose 80Hz as the default crossover for bass management in the multichannel LFE speaker application). So, placement of a bass-only speaker can be damn near anywhere in the room as long as the crossover is 12dB per octave with a -3dB point at or below 80Hz.
My bass speaker arrays meet all of the criteria I just laid out above. They are separately integrated with their own amplifiers and crossovers. They operate below 80Hz and thus do not improve most listeners' ability to recognize natural sounds or musical pitch. And, they could be placed anywhere in the room and not affect imaging or confuse the listener as to the distribution of sound. The one aspect which doesn't make them purely a "subwoofer" is that they are not a single speaker system producing a mono output from a LFE or stereo input (a fourth aspect I chose not to get into here).
Now, to be completely fair, I have argued in the past against the use of the term "subwoofer" when describing bass speakers which are clearly not operating audibly below 50Hz, or so. I recall writing book-length arguments on the S&V Forum defending the idea that all of these P.O.S. bass speakers being sold as "subwoofers" which users were bragging about owning should not be called "subwoofers" because a proper subwoofer should get lower than 30Hz. So, I get your point, but in this case I don't know what sort of bass performance I'll get below 40Hz because I am coupling 8 woofers together. I modelled the performance for one of these 10" woofers and the bass limit, which is usually defined as the -10dB point, is about 28Hz with a -3dB point of about 41Hz. That's for one woofer. My modelling software (which I developed in the early 2000's and which I trust implicitly for a single woofer's performance) doesn't have the feature to take into account coupling together more than one woofer. It can assume SPL improvements for multiple drivers, but not how the acoustic coupling might extend the bass limits. I've seen papers claiming that for every 4x increase in drivers you can expect an octave of extension, but that was written about very large line arrays used in PA systems where active DSP EQ is being used - not for small rooms like my living room.
So, I submit that my use of the term "subwoofer" is correct in this instance.
If, however, I choose to change the crossover to 200Hz, I would agree that calling them subwoofers would be inappropriate and mislead the readers of my speaker building story.
ADDENDUM:
In PA systems the term subwoofer usually referred to the speaker enclosures which have more flexibility in placement and which can be increased for a large room and reduced for a small room. Few of the pro subwoofers used in PA systems operate at all below 35Hz or so since doing so could significantly increase the likelihood of feedback and make a live sound engineer's life miserable. The term "Bass Bin" originally came from the PA and Musician world. It is often used to define the direct radiator woofer enclosures which are mated to midrange horns. It can also be used as a name for the speakers used for a bass guitar. So, in the PA world, a subwoofer typically operates from about 40Hz to anywhere from 80Hz to 150Hz. A bass bin can operate from about 40Hz up to 500Hz to 800Hz (PA) or up to 5,000Hz (Bass Guitar).