# Poisson Regression for binary outcomes - why is legitimate?

I have learned - and taught - that to build a regression model for a binary outcome one should use a logistic regression, for a outcome that has discrete counts one should use the Poisson regression, and for count data with zero inflation or under-/overdispersed data one might choose negative binomial regression instead.

Now, I have come across this thread: Poisson regression to estimate relative risk for binary outcomes, and I'm devastated. Apparently, there are quite some people who do not adhere to the rules that I have learned, and actually apply Poisson regression to binary outcomes. My world seems to become more and more complex.

As I understand, when we apply Poisson regression, we model the parameter $$\lambda$$ of the Poisson distribution via a log-linear model, i.e. $$\ln(\lambda)=\beta X$$. This makes sense if we assume that our outcome $$Y$$ follows a Poisson distribution: $$Y \sim Pois(\lambda)$$, therefore $$\lambda = E(Y|X) = e^{\beta X}$$. The Poisson distribution does not have an upper bound, therefore its pmf $$Pr(Y=k)=\frac{\lambda^k e^{-\lambda}}{k!}$$ is supported $$k \in \{0,1,...,\infty\}$$. Now, when maximising the likelihood, we cannot restrict possible values for the outcome to $$k \in \{0,1\}$$, because $$\sum_{k=0}^1 Pr(Y=k) \neq 1$$, so it would not be a valid pmf anymore.

My question is, what assumption does one make, when one (correctly) applies Poisson regression to model a binary outcome?