My generous playtesters had questions about:

• Starting Luck: with a range of 7-13, why use 3d6? Why not 1d6+6 or 1d6+7?
• Why Mana? That makes me think of either magic points, or magical sky-bread. Why not Difficulty, or Stakes for the gambling metaphor?
• It looks like learning new Approaches could be quite easy, especially if the player chooses a relatively low-Mana challenge to attempt. Is this intentional?
• Tools don't appear to have any mechanical impact aside from being used to justify new Approaches.
• The part on improving an Approach is worded a little confusingly. I think I have to (1) risk Luck at least equal to my score in the Approach, and (2) fail. So Vrunor could only improve his Invocations of Fire and Wrath Approach by making the attempt, spending at least 4 Luck, and failing the roll. Am I understanding this correctly?
• There are references to both risking Luck and spending Luck, and the rules say that if you fail your roll you lose one Luck. Does this mean you don't lose anything on a success? If that's the case, what's to stop me from just risking Vrunor's full 8 Luck on every roll? I'll be rolling 2D+12 most of the time, for an average of 19, and it sounds like there's no greater risk for risking/spending all your Luck than for risking/spending one point.
• If I make a roll without using any Luck and fail, do I still lose a point of Luck?

I updated the rules to reflect these answers:

1. 7-13 range has a different probability spread with the windowed 3d6; stacks up more at the bottom and top. Not sure I need to make this explicit.
2. Mana is a word I appropriated from Polynesian culture - as you probably know. That sin's not original to me. It's often used as a fantasy trope for "magical potency"* and indeed that's a primary usage in the original culture. But it also connotes something like "big shot," with implications for social and physical stature as well. It's a contextualized concept: you possess Mana not only as an innate quality, but also because it's accorded to you by others. There's something I'm trying to express about what makes a challenge important, and I'm not sure how to capture it, but I feel like "Mana" works for my idea in a way that Difficulty (not enough about the context of the challenge) or Stakes (too much about the situation than about the challenge) do not. Yet Mana also is not really the right word, as you point out. Perhaps "Significance" or "Stature" or "? Power" would work?
3. Yes, it's intended that characters would be easy to broaden.
4. Tools are enablers for Approaches. Without a tool, you can't use an Approach. Definitely need to clarify this.
5. I'll work to clarify this, but you interpreted it as I intended.
6. I'll get rid of "risk" - Luck is always flat-out "spent," and inevitably runs out.
7. Yes.

*Ars Magica went with vis for reasons of authenticity and non-appropriation, a sound choice.