Why does the logistic model for binary logistic regression return the probability that the outcome was in class/category 1? In the context of simple binary logistic regression (https://en.wikipedia.org/wiki/Logistic_regression) we have $p(x)=\frac{1}{ 1 + e^{\beta x}}$, where $p(x)$ is interpreted as the probability that the observation $x$ has $y=1$, i.e. $x$ belongs to category/class 1.
My question is: Why does the fitted model return the (conditional) probability for category/class 1; why not category/class 0 instead? Or differently asked: How can one determine whether the fitted model returns the (conditional) probability of category/class 1 as opposed to category/class 0?
I would like to see answers that show where in the defintion of the logistic model the assumption that $p(x)= p(y=1 \mid x)$ is made explicit.

A logistic regression proposes a conditional binomial distribution and then estimates the parameters of those conditional distributions.
Binomial distributions are defined to give the probability of class $1$ instead of class $0$.
Yes, the theory could be tweaked to do the reverse, but there is no benefit to doing so (since all of the information about the probability of being in one class is contained in the probability of being in the other class), so we stick with the convention of a binomial distribution pertaining to the probability of class $1$ instead of class $0$.
