I thought this might have come to an end when I agreed to disagree, but ok, let's continue!
I don't do that (agreeing to disagree). It simply signals a popcorn break.
Let me ask you some questions:
a) What happens to the control of the aircraft at Vmc?
In question a, it doesn't matter which Vmc we are talking about, because the answer is the same regardless: the answer is that control is lost.
Well, not really. It's the point at which control can no longer be maintained, as defined by a 20 degree change in heading. A loss of control is a different matter entirely. That might sound like semantics, but it's not.
b) Can you explain why Vmc is lower when 5 degrees of bank towards the live engine is used compared to when the wings are level
In question b, the context tells us that we can't be refering to Vmca, we must be talking about the actual speed at which control is lost. If we were talking about Vmca, the question would be nonsense because Vmca is defined as not being wings level, but the question asks about Vmc with wings level and implies that Vmc exists, and is different, if wings are level.
Sure we're referring to Vmca. I don't think you understand what Vmca is, yet.
You still seem to think it represents something other than what it does. Rather than explain it again (I think it's been very thoroughly covered thus far, insofar as the terminology goes), read again.
Where do you find Vmca defined as wings not level? Vmc is Vmca. Vmca is Vmc. Vmca is Vmc in the air. Remember?
As to your question, the five degree bank isn't used with all aircraft. When you move up to transport category aircraft, for example, you'll find that wings-level is the order of the day when addressing an engine-out, in most cases.
In both of these cases, the phrase "Vmc" is absolutely adequate.
It's adequate insofar as the fact that directional control can no longer be maintained. In the context of the red radial line with reference to Vmc or Vmca, specific conditions are imbued and inferred. With respect to the actual point at which directional control can no longer be maintained (nearly always something other than the little red radial line, both Vmc and Vmca directly adress that speed. It's the point at which directional control can no longer be maintained, in the air.
Vmc, of course, can also refer to the same thing on the ground, which is Vmcg, or in approach configuration, which is Vmcl, or Vmcl-2.
c) Does Vmc change with height?
But in question c, it is not. The answer to the question depends on what Vmc we are talking about. If we were being asked about Vmca, we might answer "No, Vmca is always quoted for sea level". If we were being asked about the actual speed where control is lost, we might answer "Yes, Vmc decreases with height in an aircraft with normally-aspirated engines because the good engine produces less power the higher it is."
So, in the case of question c, the phrase "Vmc" is ambiguous.
Perhaps you mean to ask if it changes with altitude, and with most light airplanes, yes, it does. Vmc is Vmca, and Vmc and Vmca change with altitude, though in most cases the published numbers do not, nor does the red radial line move.
For certification purposes, the number tends to remain fixed, but the relationship to actual directional control varies with the aircraft and powerplant system in use. It's fair to say that EAS and TAS for Vmc changes with altitude, though you may or may not see an actual change in the indicated airspeed at which directional control can no longer be maintained. You haven't provided enough information.
Where Vmc, meaning the ability to no longer maintain directional control, changes with altitude, the term is not at all ambiguous, because it means just what it says: the ability to no longer maintain directional control.
The numbers may be different, but the term is still valid and precise.
Re-read again. You're missing something. Can you figure out what it is?
You may have to have the conversation alone for a few days. Time to go fly.