Asking people to self-select a single question is not going to give you much signal. Why not use an established test like political compass? - https://www.politicalcompass.org/test
The most important thing to remember in all such engagements is that self-reported values are never reliable. People are always subconsciously biased to report themselves as more like what they want to be, and what they want to think of them as. A real evaluation of their place on the political spectrum would need to be compiled from data on their history/expressed opinions.
Though it'd be difficult, an interesting potential use of a site like this is helping people realize that they are more left or right leaning than they previously thought. You'd have to be careful about the messaging since a lot of people would be offended if the computer told them "I know you said you're a lifelong Democrat, but we think you agree with Trump 75% of the time". But if it's done right, this type of tool could open a lot of minds and neutralize the poisonous, vindictive political discourse that people often use to pit groups against one another.
From the "About Us" page (http://www.yourmorals.org/aboutus.php):
This website is a collaboration among social psychologists who study morality and politics. Our goal was to create a site that would be useful and interesting to users, particularly ethics classes and seminars, and that would also allow us to test a variety of theories about moral psychology. One of our main goals is to foster understanding across the political spectrum. Almost everyone cares about morality, and we want to understand --and to help others understand -- the many different ways that people care.
However, as many commenters have pointed out, the current design makes it somewhat difficult for people without strong partisan affiliations to choose a side. I'm going to look into some sort of short quiz like the political compass that might make signups easier for people (and also increase the signal, as you mentioned).
Maybe I'm missing something, but that's basically all we have right now. Is the goal to get people to be tribal and emotional in the same place instead of apart?
Also, I have strong affiliations, especially on particular topics, but I can't really identify as right if Trump is "far right" or left if Warren is "far left". It doesn't make me confused about who I am. It's just that people aren't even measuring right or left on the same axes.
Your beliefs notwithstanding, I think that there are a fair number of people who do identify as either unambiguously right-leaning or left-leaning, and I intended the blue and red categories on the site to be for them. I was hoping that people like yourself that don't feel comfortable with either label would be comfortable choosing "neither". It's meant to include both people who are confused about their stances, as well as people who are sure but don't identify with either the right or the left.
However, it looks like people didn't interpret the categories the way that I expected them to. I'm going to rework the signup page to try and make the third category more appealing to people like yourself (and like myself!) who have strong views but don't see themselves on either the left or the right.
I don't identify myself as "independent", so I suspect I won't really be served by a grab bag group (1). To put it another way, I suspect politics is becoming increasingly "long tail", so I suspect that even three categories will have issues.
Though in a long tail world, maybe there's a niche for that even.