Lotteries = probability distribution? Are &quot;lotteries&quot; in the model for choice under uncertainty not just probability distributions?
Yes. See for example this passage from Choice under Uncertainty by Jonathan Levin:

2.1  Prizes and Lotteries
The starting point for the model is a set $X$ of possible prizes or consequences. In many economic problems (and for much of this class), $X$ will be a set of monetary payoffs. But it need not be. If we are considering who will win Big Game this year, the set of consequences might be:
$$X=\lbrace\text{Stanford wins}, \text{Cal wins}, \text{Tie}\rbrace.$$
We represent an uncertain prospect as a lottery or probability distribution over the prize space. For instance, the prospect that Stanford and Cal are equally likely to win Big Game can be written as $p=(1/2,1/2,0)$. Perhaps these probabilities depend on who Stanford starts at quarterback. In that case, there might be two prospects: $p$ or $p'=(5/9,3/9,1/9)$, depending on who starts.

(not sure what I was thinking yesterday evening. It seems like I completely misunderstood the question)