I will bear in mind the FAA's loose definition of Vmc in future!
It's not a "loose definition." It's quite precise.
Vmc means the speed at which directional control can no longer be maintained, period.
What Vmc doesn't tell you is when and where.
What Vmc does not do is give you specifics. Vmc in the air is not the same as Vmc on the ground. Vmc in the air under the conditions predicated for Vmca is not the same as Vmc while configured for landing, Vmcl (or Vmcl-2, as applicable).
To assign the red radial line on the airplane as Vmc is correct, because that's the published speed at which directional control may be anticipated to be no longer available under the prescribed certification conditions for air operations. Vmcg is not shown on airspeed indicators, nor is Vmcl. therefore, the red line is Vmc, one of several Vmc's but it is specifically Vmca. To refer to it as either one is correct, because this is a correct definition. Not a "loose" one. Vmc on the airspeed indicator is the published certification number at which directional control can no longer be maintained within 20 degrees when the critical engine is suddenly retarded, the good engine maintained at takeoff power, an aft center of gravity with the least favorable weight, in the takeoff configuration. This is certification Vmc, and as it's an airborne number in the takeoff configuration, it is also Vmca; the terms are synonymous for this application.
The same cannot be said of the red radial line and Vmcg, Vmcl, or Vmcl-2. Those numbers are not the same as Vmca (although they are Vmc numbers). Accordingly, whereas we know that the red radial line speaks to specific conditions of establishment, we know that this particular Vmc value is only an airborne, takeoff-configuration number. It is Vmca, but it a Vmc value none the less. To refer to any of the Vmc values as Vmc is correct, but to refer to them in context is more correct, or more precisely, more precise. That is, while Vmca, Vmcg, Vmcl, and Vmcl-2 are all Vmc, a more exact rendering of "Vmc" when referring to these speeds appends the letter or letters which speak to the circumstances under which the speed is derived.
It is not correct to say that Vmc is the red radial line, but that Vmca is the actual speed at which control will be lost. Neither is it correct to say that Vmca is the red radial line, and Vmc is the actual speed at which control will be lost. Vmca is Vmc. Vmc is Vmca. Vmca is Vmc in the air. Vmc in the air is Vmca. This isn't a loose definition; it's a regulatory one, a design and standards one, and one spelled out throughout technical manuals, handbooks, manufacturer publications, certification standards, and so forth. It's not the definition which is loose in this case, but perhaps your understanding of the subject. I realize that you may have been taught differently in class, but that doesn't change the fact.
As Tinstaafl noted, whether one chooses to refer to the red radial line (or published numbers) as Vmc or Vmca makes no matter. It's a general guideline. The actual airspeed under which control might be no longer possible to maintain will vary with aircraft loading, configuration, ambient conditions, center of gravity, and so forth. The actual Vmc (or Vmca, if you will) varies, and can be lesser or greater than published.