Hingeyness refers to a time in which we have an unusually high amount of influence over the future of civilization, compared to people who lived in the eras before and after. Let's look at a simplified model of time. In this model are only two possible choices per year. The number inside the circle refers to the amount of utility in that year and the two lines are the two options that this year has to decide on. The amount of utility which each option will add to the next year is written next to the lines.
If you live in the year of 0 utility, the previous year will seem to have been very hingey and the year before that not so much. Even if the 0 thinks that the 1 made the right choice (because it added more utility to the world), we can see (with our godly knowledge of all possible timelines) that the better choice would've been the +1 choice (because you get a much more favorable choice next year).
All older years have more ripple effects, but does that make 1 more hingey? The difference in utility of the choice 3 had to make is bigger than the difference in utility the 2 and 1 had to make. But wouldn't you rather end up like 2, with a choice that doesn't make much of a difference but has huge good outcomes either way? The absolute utility that 1 and 2 could add are the same, but the relative utility is very different. So, what is more important for the hingeyness? Which of these three times is the most hingey and why?
We cannot do experiments on history, so I'm not sure it is even possible for 0 to know that 8 was a possibility. And in our world the amount of branches are infinite, making this whole discussion even more difficult. When we are looking at the potential branches in the future, should you make the choice that will lead you to the cluster of outcomes with the highest average utility or to the cluster with the highest possible utility? Here I am pretending there even is such a thing as an outcome. I would argue the 1-3-6 timeline is just as good as the 1-2-7 timeline, since they both have equal utility over their timespan. In reality the timeline will continue onwards so choosing timelines that stretch on for longer is in most cases better (unless you're a negative utilitarian).
I could go on adding complexity, but I think you can see why I question if hingeyness is even a coherent or measurable concept.
I feel like I should clarify what I mean with "absolute hingeyness" versus "relative hingeyness". Say, we have a first choice between adding +1 utility and +2 utility and then a second choice between adding +4 utility and +6 utility. The difference in absolute utility is +1 for the first choice and +2 for the second choice, which gives the second choice more "absolute hingeyness". +2 is twice as large as +1 compared to +6 being only 1.5 times larger than +4. This gives the first choice more "relative hingeyness".
If you have more options with more possible futures you can even start thinking about mean vs mode vs median utility.
I really like this model and will probably use it to think about hingeyness quite a lot now!
I'll make an attempt to give my idea of hingeyness, my guess is that the hingeyness is new enough an idea that there isn't really a correct answer out there.
You can think of every choice in this model as changing the distribution of future utilities (not just at the next time step but the sum across all time). Hingier choices are choices which change this distribution more than any other. For example a choice where one future branch includes -1000 and a bunch of 0s and the other includes 1000 and a bunch of 0s is really hingey as it changes the portfolio from [-1000, many 0s, 1000] to [-1000, many 0s] or [many 0s, 1000]. A choice between [many 0s, 10] and [more 0s, another 10], is not hingey at all and has no effect on history. A good rule of thumb is to think of a choice as hingier the more it reduces the range of possible utilities.
As an extreme example, choosing between powerful world governments where one values utopias and the other values torture seems very hingey as before we had a large range of possible futures and after we're irreversibly in a really positive or negative state.
I'll apply this model to some of your questions below.
All older years have more ripple effects, but does that make 1 more hingey?
I think in the diagram above 1 is coincidentally quite hingey because you choose between a world where you're guaranteed a 7 or 8 utility down the line or a world where you have a range between 0 or 6. The range of possible choices for one option is very different than the range for the other. You can imagine a similar timeline where the hingy moments are somewhere else (I've done an ugly drawing of one such world) As you can see in this timeline choice 1 doesn't matter at all in the long run because the full range of final options is still open to you but the second tier choices (and the one labelled 0 in the third tier) matter a lot as once you make them your range changes in big ways.
The absolute utility that 1 and 2 could add are the same, but the relative utility is very different. So, what is more important for the hingeyness?
I think neither are the key values here. Hingeyness is about leverage over the whole human trajectory so the immediate changes in utility are not the only thing we should consider. We care more about how this affects aggregate expected utility over all the remaining future states. This is why irreversible choices seem so concerning.
One last thought here is that hingeyness should probably also include some measure of tractability. It could be one choice has a large effect on the future but we don't have much of a capacity to affect that choice. For example, if we discovered an asteroid heading towards earth which we couldn't stop. There's no point in considering something the hinge of history if we can't operate it! Currently, I don't think that's in the model but maybe you could add it by imposing costs on each choice? My guess is this model could become pretty mathematically rigorous and useful for thinking about hingeyness.
(I'm only addressing a small part of your question, not the main question)
When we are looking at the potential branches in the future, should you make the choice that will lead you to the cluster of outcomes with the highest average utility or to the cluster with the highest possible utility?
I'd say the one with the highest average utility if they are all equally likely. Basically, go with the one with the highest expected value.