RAW in Shadowrun has never been a simple matter, and quite frankly a lot of things even Rules as Written has a lot of room for interpretation and sometimes entire rules areas where things just aren't defined.
So yes, there is a difference between some (rare) qualities or items that remove or heal damage, and qualities or items that reduce damage taken. The verbage that they use for those two different operations are already quite different. Your comment that the use of "incoming" was a clarification on that difference is what I was trying to refute.
The SR rules are filled with terminology usage that is never clearly defined. In fact, there is exactly 1 instance of the core rulebook referring to damage as "incoming damage" which is on page 397 in the Example for Hardened Armor. It is referring to the damage when it is being compared against the Armor Value. The only other time they even use the term "incoming" is in reference to attacks coming at a target. All of these uses are outside attacks that are inbound toward the target.
I've always viewed Drain as the Magician's ability to channel the energy that they are shaping into the spells they are creating. As they channel more magic, the strain becomes greater, resulting in a higher amount of Drain to resist. This is supported by the Sensitive System negative quality which outright states that because of your more sensitive system you sometimes take more damage as the energy being channeled hits your body harder. So I relate this to essentially Magical Fatigue Tests. I wouldn't count Fatigue as "incoming" damage either, it is damage caused by overexertion (like Running) or lack of something your body otherwise needs (Hunger, Thirst, Sleep, etc).
And yes, the mana is certainly coming from inside the mage or at least channeling through them. The manasphere isn't "taking a spell order" dishing out the effects of the spell to the target and then giving the mage back their change in the form of Drain. There isn't some outside source sending the Drain at them independent of the Magician.
With the Elemental Focus quality, the Magician has essentially tapped into a way to increase their control over their chosen element (dice pool bonus), but at the cost of channeling part of that damaging effect through themselves. Essentially changing the Drain Value to result in the effects of Elemental Damage. The Elemental Mastery quality protects you (and your gear) against incoming threats of that Element, but (to use a Fire reference), it doesn't help you when you're burning yourself from the inside out. You handle the elemental effects better (being immune to the secondary effects), I just don't think it reduces the actual Drain Value. Oh, plus the fact that Elemental Focus states that the drain is "taken" as that Element, which would be after it is "incoming." Just noticed that part...