Let $A_1, \dots, A_n$ denote a random sample from the probability mass function
$$P(A = 1) = \dfrac{1}{\alpha + 1}, \\ P(A = 2) = \dfrac{\alpha}{\alpha + 1},$$
where $\alpha > 0$.
My understanding is that the $A_i$ are Bernoulli random variables with probability of success $\dfrac{\alpha}{\alpha + 1}$. So if we now define a new random variable $B_i = A_i - 1$, then wouldn't this also be a Bernoulli random, but instead with probability of success $\dfrac{1}{\alpha + 1}$?
Bernoulli random variables traditionally have the values $1$ or $0$, so the $B_i\in \{0,1\}$ would be Bernoulli but the $A_i\in \{1,2\}$ would not be.
Any discrete random variable with only two outcomes ("binary-valued") is a linear transformation of a Bernoulli random variable: for instance, $A_i=B_i+1$ where $B_i$ is Bernoulli. If the two values are $a$, with probability $p$, and $b$, with probability $1-p$, then $(a-b)\text{Bernoulli}(p)+b$ will do the trick.