That ratio seems to say, that “hue begins after white”?
That is, when the common part (white) is removed.
I’m not entirely sure what that means, but I think the answer is “yes” for (I stress) the simple arithmetical systems. If we add the same constant k to all three channels (k can be negative), the ratio becomes:
( (mid + k) - (min + k) )
-------------------------
( (max + k) - (min + k) )
(mid + k - min - k)
= -------------------
(max + k - min - k)
(mid - min)
= -----------
(max - min)
… so the ratio is unchanged, so the hue is unchanged.
Given the two conditions I mentioned above, the first condition tells us which of the six segments in the hue circle we are in. The boundaries of the segments are: red, yellow, green, cyan, blue, magenta and back to red. For example, if red is the maximum channel and green is the middle channel, the hue is somewhere in the segment between red and yellow.
Suppose RGB values (R,G,B) are in the range [0,1]. We can subtract the minimum value from all three channels without changing hue. So now the mimimum is zero. Then we can divide all channels by the maximum without changing hue, so the maximum is now at 1.0. In our example, the blue channel is zero, and the red channel is one, and green is somewhere between zero and one: the RGB values are (1, G’, 0).
The middle channel then gives the angular distance of this hue to one of the boundaries. If green is zero, the hue is red. If green is one, the hue is yellow. For other values of green, the hue is orange or reddish-yellow or yellowish-red or whatever.
Since you have much reading to do, I won’t offer you new resources. I just want to caution that terminology can be mixed or used to mean different things in the literature. Be careful not to assume meaning because the author(s) may be using their words differently. There are also mistakes and laziness (as in the authors don’t actually know what they are talking about). This happens in research papers much more often than you would think! That is why it is better to discuss what you have learned with other people, e.g., on this forum, to verify whether you and the sources you have read are indeed correct.