I found that there are, so to say, two (at least) types of covering. I>call them horizontal and vertical. The horizontal type is when one uses smile and tension of the facial skin to put the sound into the mask. This way the ring in the sound becomes very strong. Sometimes people raise upper lip to achieve greater tension of the skin. This way of covering is suitable for>tenors, and I saw many of them to do this way. The other type of covering is vertical - when mouse is opened vertically in O form. Usually one tries to keep his chest sound when doing such a covering. This type of covering is suitable for baritones and basses. What you say about mouth positions is correct but I would not call them different kinds of covering.
I think that there is basically only one kind of covering but various ways of doing it. Covering is a matter of adjustment or tuning to achieve proper resonance without stressing the voice. Many male singers cover with a narrow, vertical mouth position. Corelli opened his mouth very wide up and down but also side to side, not nearly as narrow as Bergonzi, for example. Gedda and Kraus achieved excellent results with a smiling position. Perhaps this mouth position is good for notes above B flat. Tucker used a very small mouth position even for his highest notes. The crucial thing, I believe, is strong breath support and relaxation in the mouth and throat. Beyond this, different singers may find various mouth positions which work for them. I am a tenor and cannot sing high notes with the narrow mouth of Bergonzi or the small mouth of Tucker. Trying this makes my throat tighten. I sing high notes best with a position similar to Corelli's: vertical, yes, but with some widening side to side. This works best with my jaw, muscles, etc. I am an older singer but I have a good high B flat and can sing high C. I have just been pounding out high B flats in chorus rehearsals for our Aida performances in October. Use the best means to achieve the result.