I think willpower and akrasia are both kind of fake concepts. Here's a different model: you have a bunch of parts, approximately in the IFS sense. Those parts want different things, and sometimes they come into conflict. Different parts are in control of what you actually do at different times. Some of your parts are verbal and some are not. (I don't think "verbal = S2, nonverbal = S1" is necessarily a helpful frame here; for example, focusing seems to be related to nonverbal parts, but it sort of breaks the S1/S2 dichotomy by being nonverbal but slow.)
"Akrasia" is when a nonverbal part is in control (and wants to, say, watch TV) and a verbal part wants to do something else, and says so. "Willpower" is when a verbal part attempts to forcibly override a nonverbal part. The skill to try to build is understanding what your nonverbal parts actually want, facilitating communication between parts, and either making compromises between your parts or getting your parts to agree on what to do (I'm genuinely uncertain which is better).
Edit: Also, my impression is that this is essentially the model currently being taught at modern CFAR workshops, although people who have been to a CFAR workshop more recently than me should chime in about whether they agree or not.
The skill to try to build is understanding what your nonverbal parts actually want
Isn't that the easy part? Just look at what it's doing: if I'm eating a bag of chips instead of working out than it means that my non-verbal part wants to eat a bag of chips. Or there's something else?
Rationality is designed to make you win, to help you attain your objectives. One of the most prominent phenomena getting in the way is akrasia, the lack of willpower preventing us to perform whatever action we want to do.
So, wait, do we want to act or not? There are two facets of willpower, which I'll describe using Kahneman's System 1 and System 2, the two working modes of the mind (bear in mind it's a simplification to get the point across).
What we call System 1 willpower is what gets you up in the morning, out of a good shower, what makes you start things, the grit, the sudden confidence, the "just do it" part. It's the short-term impulse of "I can go one step further and my body shall obey. It's the thing many productivity methods want to trigger. For example, the 2-minute rule ("if it takes less than 2 minutes to perform, do it now") is based on the intuition that you won't need a lot of effort to get something done, akin to going briefly out of your way to help a friend. Of course you can do it. No planning needed. It's very appealing to your brain.
What we call System 2 willpower is the high-level justification, the reason behind the actions, the thought-out plan, the will to make good decisions, to make things right, to keep akrasia in check in order to achieve long-term goals, the thing that sees problems as obstacles to be overcome instead of monsters to avoid. It's the background buzzing of "I want to go there and I shall find a path". It's the thing many productivity methods want to manage. The Getting Things Done methods starts with a list of tasks, which you can process in orderly manner, setting long and short-term goals and stay in control at all times. It's also very appealing to your brain.
However, what makes you actually do things is S1 willpower, not S2. S1 is more low-level, and has no knowledge of plans. Your willpower can be misguided. You want to throw paper planes, scribbling randomly, following click-bait headlines, when your plan is to work, or train, or be otherwise productive. S2 willpower is supposed to keep S1 in check, to make you perform counter-intuitive actions for your own good.
The willpower tricks work, but they're costly. They require a part of your attention, a conscious effort to keep yourself focused. It's just much less costly to have S1 willpower run everything, with S2 staying where it should belong: abstract thought, not micromanagement of S1. This is flow, where S1 willpower remains high and has no reason to go down because everything goes smoothly (and it's a fragile state, any disruption can get you out of it).
Conversely, ignoring completely S1 willpower and forcing yourself to do things isn't safe nor productive. If you ignore the natural drive that comes from pleasant activity, you may coerce your will into whatever "optimal" process you designed, working yourself ragged. You're not a machine. Your body is part of you, not a weak container, nor a lazy-by-default entity you should negotiate with.
My model of effective anti-akrasia isn't to poke System 1 repeatedly but to engineer situations where S1 agrees with S2 on what is best for you; where you don't encounter contradicting signals of "I don't really want to do that, is that necessary?" all the time. The goal is to make productivity the default state, to make it more attractive than doing nothing. How we can do this will be covered in later posts.
Special thanks to regex, who sparked the discussion leading to this post. I borrowed his "more attractive than doing nothing". It's a fascinating concept.