I can see three basic types - from easiest to most difficult. Let's start with the easy first.
1. First order/No Time-Phase coherence: this speaker will use first electrical order, but there is no time-phase coherence. It will not be able to produce a proper step response. It's more or less conventional with the exception that it uses first order filters.
2. First order/Time-Phase Coherence, BUT NO "Time coincidence"
(which will be explained in #3).
This speaker will be able to produce a proper step response, BUT and an important BUT. It may not be able to produce an excess phase of 0 degree from say 50Hz to 20KHz. This means that the speaker, for example, may have a phase shift of 50 degree or more or could be a full 360 degree at 15KHz, but only 5 deg at 500Hz. That is its excess phase will vary especially at higher frequencies as the tweeter approaching 20KHz. John Atkinson would agree this speaker meets his definition of "Time-Phase Coherence" since it could produce a proper step response. My guess is most speakers that were measured by John Atkinson would fall into this category. I've seen some measurements done on Vandersteens speakers and I was like ... hmmm... I am not quite sure. But the most stringent definition is reserved for #3.
3. First order/Time-Phase Coherence AND Time-coincident: this is the most difficult definition for any speaker to meet. That is it has to be able to produce a proper step response like in #2, BUT it also has to be able to have a 0 degree of excess phase from DC - 20KHz. In reality, no speaker will have absolutely 0 degree, but the variations should be very small. I believe Thiel claims that their speakers excess phase shift is only a few degrees (less than ten). To be honest, I am not sure many speakers in the entire history can meet this definition.
So to summarize, you have three distinct possibilities from easiest to hardest:
1. First order/No Time-Phase coherence
2. First order/Time-Phase coherence BUT NO "Time coincident"
3. First order/Time-Phase coherence AND "Time coincident"
As for terminology, I guess you can call anything you want, but as far as the measurements, those are the three categories.
With real data, graphs, plots, simulations those three cases can be clearly demonstrated. Just talking about it making things more confusing.