All of that seems accurate. The companion did indeed win a skirmish against that minion, and the minion lost to that companion.
No, as you can see in the detailed step-by-step explanation, the Companion did win a skirmish (step 7.4.1), but not "involving [quality]", as the minion was not in the skirmish at this point, and that's a definition of "involving". So you wouldn't be able to draw a card in this case (of one minion).
So to summarize:
If you follow your way of thinking, where minion "loses" mid-skirmish if removed from skirmish, you wouldn't be able to draw a card, and if there was more than 1 minion you would be able to draw a card.
If you follow my way of thinking, where when skirmish resolves game decides which characters are winning (=on the winning side and in skirmish when skirmish resolving) and losing (=on the losing side and in skirmish when skirmish resolving + all removed characters), you'd be able to draw a card in both cases.