What is the intuition behind the odds scale? What is an intuitive explanation of the odds scale?
In a logistic regression such as $$logit(p) = \beta_0 + \beta_1 x$$
we often interpret $\beta_1$ by looking at the odds ratio, $e^{\beta_1}$, which has the
interpretation that a unit increase in $x$ is associated with a change in the odds of "success" by a factor of $e^{\beta_1}$.
Say I'm making basketball shots, and my successful shots are well modeled by a logistic regression $logit(p) = \beta_0 + \beta_1 x$ where $x$ is meters from the basket. If $e^{\beta_1} = 0.5$, then each meter that I step farther from the basket halves my odds of making the shot. This "sounds" fine, but I don't have have an intuition about what halving or doubling my odds means.
I thought of one interpretation of odds, which is the following: my racehorse is in a race with 9 other horses, and all 10 are of equal ability. So each has odds of 1:9 of winning. Then one way of thinking about the odds ratio is that halving my odds, or doubling my odds-against, is like doubling the number of opposing horses to 18.
In site searches I haven't found any intuitive interpretation: here says it's not intuitive, and here suggests that when people say "twice as likely" they aren't clear which scale is being used.

In the frequency interpretation, probability is the number of successful shots divided into the total number of shots (at each distance $x$). The odds is the number of successful shots per failure. That seems to be an intuitive description!
So, in your example, 1 meter longer from the basket, the odds is halved. So if the number of failures is the same as before, the number of successful shots is half.
In your other, horses, example, your interpretation seems fine.
