At the risk of being a bit repetitive from my earlier comments, I believe there are a few notable caveats and assumptions made in my answer. Wherever possible, I’ll try to highlight the assumptions made, and how they impact my opinions. That said, any clarification you can provide regarding what exactly your project’s scope and goals are, the more I can refine some of the literature suggestions and my takes (for what that’s worth- admittedly, not a whole lot haha).
The first major point to make is that there really is a lot of variation in terminology used inside the field of decision theory. That’s partially driven by the different underlying axioms different models of utility are based on. Using a relatively well-known example, Savage’s axioms don’t assume objective probabilities need to exist for most events. This contrasts with Expected Utility theory, for example, which does directly require some notion of objective probability (whether it’s known or not) of an event’s likelihood of occurring. Therefore, if your model/paper is assuming Savage’s axioms, definitions of uncertainty and risk won’t necessarily be tied to the nuances of how well known a probability is (it allows a bit more flexibility at the real ground level). At the other extreme, Expected Utility Theory, because of the strict reliance on probabilities from the Independence axiom, assumes there’s no real difference between a known and unknown probability- through reduction, any multi-stage uncertainty collapses to a single dimension of uncertainty.
I don’t mean to get too far removed from the subject at hand, but I feel it’s worth keeping in mind the high level of context dependence the definitions of these terms often have- “risk” can mean something to Savage, but something slightly different to von Neumann, and something else to Kahneman and Tversky.
The next related caveat on this terminology is in regards to the fluidity of definitions over time. Even beyond the different implications the terms have for different utility structures, different authors have used these terms in a variety of related, but subtly different ways. I think your definitions fall reasonably well inside the umbrella definitions for these terms. As long as you explicitly define how you’re going to use words like uncertainty, risk, and ambiguity, I think you should be fine.
**EDIT: TO CLARIFY A BIT MORE ON THIS POINT: Historically, several works have directly conflated ambiguity and uncertainty, defining them as the same concept, contrasted with risk. While this has fallen a bit out of favor it seems (using the ambiguity/risk phrasing more commonly), it would still be a historically correct definition. Hence, your definitional example you provide in the question is correct, but perhaps a bit dated. That said, here's an example from 2012 that does include the duality explicitly in their definition:
In real life, of course, most important decisions present a mixture of risk and uncertainty. Since Keynes (1921), Knight (1921) and Ellsberg (1961) distinguished between risk (known probabilities) and uncertainty/ambiguity (unknown probabilities), there have been many studies of the differences between risk and ambiguity attitudes.
Though again, this paper is a bit more outside the central decision theory core (applying the field to healthcare and medicine).
The last major source of terminology variation I’ve noticed is regarding what the format of the project is. For example, elite econ journal articles published recently (a few examples provided below) tend to hew more closely with the emerging consensus definition in recent decision theory work in risk and ambiguity attitudes, and the theoretical implications. On the other hand, if your project is for lecture notes for a class, or a book chapter, then you get much more leeway in how you delineate between these similar concepts- again as long as you’re clear at the beginning. If you’re writing for a journal article in a related field, like finance or management science, then there very well be conventions in those journals that don’t conform to my experience. I speak from a pretty pure economic theory background, and while I haven’t read everything out there on the subject clearly, I’ve spent a fair amount of time working with a decent subset.
So What are the proper definitions in my experience? Traditionally, you can consider one way of partitioning the set of all decision models into: decisions with uncertainty as problems where not all outcomes are necessarily known, and decisions with no uncertainty where all outcomes are deterministically known (and there are no “economic lotteries” in any stage). While there are of course many additional ways of partitioning these two subsets of problems, you can consider problems with uncertainty as being composed of:
i)problems where probabilities of each outcome is known to the agent as models of risk, and
ii)problems where some probabilities are not point identified for the agent, which makes up models with ambiguity.
Again, of course, these two subsets don’t always make intuitive sense- if your agents all follow expected utility theory, then there’s no difference between (i) and (ii) from a utility perspective. Any uncertainty over the probabilities (i.e. ambiguity) the agent faces collapses into just a compound lottery (i.e. a model of risk). This, for example, is the issue Ellsburg Paradox “exploits” to show an intuitive failure of expected utility theory.
The “modern” definitions draw from Ellsburg’s 1961 landmark paper (starting on page 657), though it should be noted that here the notions of “ambiguity” and “risk” aren’t particularly clearly delineated, they’re defined as I described above. Other utility models started considering “ambiguity” as it’s own, independent feature of preferences largely starting in the late 1980’s. The definition of “ambiguity” becomes a bit clearer in Segal 1987, starting on page 176, then continuing to 177
Ambiguous probabilities (i.e., situations where decision makers do not know the exact values of the probabilities) has some clear economic relevance… Other situations where ambiguity of the probabilities may play a significant role occur in search problems or in optimal investment problems. In all these cases, decision makers have some information about the objective probabilities, but they do not know their exact values. This paper suggests that the ambiguous lottery (x, S; 0, S) (ambiguous in the sense that the decision maker does not know the probability of S) should be considered a two-stage lottery, where the first, imaginary, stage is over the possible values of the probability of S.
(And is then defined in far more precise detail starting on page 183.)
Another prominent early example of adding this sensitivity to ambiguity conceptually into the modeling framework can be found in Segal 1988 , which implicitly uses the definitions for “ambiguity” and “risk” as I defined above.
More recent papers traditionally shorten their definitions to a common theme. Using a relatively random smattering of papers I’ve seen recently, this 2011 working paper defines the terms:
developments in the theory of decision making under ambiguity (i.e., subjective uncertainty about probabilities) recognize that ambiguity is not always treated the same as a known risk
This 2014 Handbook chapter similarly defines decision making under risk and ambiguity:
In many decisions under uncertainty, the decision maker has only vague information about the probabilities of potential outcomes of her actions. Following Ellsberg (1961), such situations with unknown or uncertain probabilities are often called ambiguous, to distinguish them from situations with objectively known probabilities, which are typically called risky.
Recent experimental papers, attempting to measure attitudes (usually in different specific settings) toward risk and ambiguity, have also followed in this general definition trend. For example, this work in Econometrica in 2017 measuring the effects of different types of ambiguity on observed decision making highlights the difference between “compound risk” (just a set of two lotteries with known probabilities of winning in a row) and “ambiguity” (where one stage led to uncertainty over the true probability of winning in that stage, which varies exogenously and unknown to the individual). For example, an example of a lottery with compound risk is:
Stage 1: flip a fair coin. If it’s heads, then go to Stage 2a. If it’s tails, go to step 2b
Stage 2a. Draw a card from a perfectly shuffled standard deck of 32 cards. If the card is not a spade, then win \$100. Else, win \$0
Stage 2b. Draw a card from a perfectly shuffled standard 52 card deck. If the card is a spade, then win \$100. Else, win \$0.
As you can see, while the probabilities might be dependent on previous draws, the objective likelihood is known at all stages. An example of an “ambiguous lottery,” then could be:
Stage 0: Draw a number $n$ randomly between $0$ and $50$, but do not show the individual playing the lottery. Then define the number of red balls in an urn as $25+n$, and the number of yellow balls in the urn as $75-n$.
Stage 1: Draw a ball randomly from the urn and flip a fair coin. If the result is either $(red, heads)$ or $(yellow, tails)$, then win \$100. Else, win \$0.
Note that, of course, in both cases the overall probability of winning is 50/50 in either lottery. However, sometimes we find that individuals will prefer one lottery over the other, suggesting preferences are partially formed along some other dimension.
Sorry this has dragged on. I’ll get back to summarize and add a few more references in a bit, but hopefully this can start the conversation going a bit. If you have any details about the project you have in mind, then I can try and tailor what I add!