I would say the definition everybody else has been using for several decades is "covers the range below the woofer on the speaker you actually have." Using your definition, the 10s of millions of pro audio subs sold over the years were all named incorrectly. The driver manufacturers all named the drivers incorrectly, all those THX guys need to find a dictionary....You seem to have a strange definition of "true sub-woofer"... strictly speaking a sub-woofer covers the range below the woofer on a full range speaker, and all full range speakers I have ever heard of go down well below 65Hz. (around 30Hz is typical for a full range... so a full octave below 65Hz)
In the title of the thread was "big speakers," you have many in the thread throwing smaller towers into that group because they're much bigger than bookshelves. You seem to have inserted "full range" into the conversation. I don't think anybody would disagree that true full range towers, if placement/DSP allowed them to be free of room/SBIR nulls in the bass will do just fine without a sub. At least in a small room. But I think the number of true full range towers that can fill a room of any decent size without running into distortion serious distortion/compression in the lower frequencies are pretty few and far between. Case in point:
I think anybody expecting two 6 1/2" woofers to play bass with any authority at 30, 40 maybe even 50 Hz is going to be very disappointed unless it's strictly small room/low volume.The Kef R7 Meta have an in-room response down to 27Hz (-6db)
I don't think anybody who can afford true full range towers is going to think mating them with a $300 sub will be an improvement. There are plenty of very affordable subs that will easily surpass the capability of the JBL M2 at low frequencies. A lot of M2 users also use subs...Look at a typical mass market subwoofer - eg: Klipsch R-100SW.