How do you specify regression models My question is about the nomenclature of specifying models. I have a specific example, taken from a paper:

The outcome $Y$ is an independent Poisson variable, with means determined
  by the parameters $\alpha$ and $\beta$, such that
  $$ Y \sim \text{Poisson}(\alpha\beta)$$

Q1) What is the meaning of this statement? From what I understand, the distribution of Poisson variable is given by

with mean equal to $\lambda$. Thus, from the specification above, is $\lambda = \alpha\beta$?
Q2) if I wanted $\lambda = \alpha + \beta$, would the model specification then be:
$$ Y \sim \text{Poisson}(\alpha + \beta)$$
Q3) How would one specify logistic regression models and linear models in this manner? Particularly for logistic models I have seen
$$\text{logit}(\text{Pr}(Y = 1)) = \beta_0 + \beta_1 x_1 + \dotsm$$
How is this converted into the specification format given above?
 A: This can all be made rigorous by thinking about a "mean model". Poisson regression has the following mean model:
$$ \log E[Y|X] = \alpha + \beta X$$
The log transformation of the expected response is called a "link" function in a GLM.
q1. yes that's basically it. And a Poisson regression model estimates a linear model for the log-transformed response, (a GLM with a log link) so the $\alpha\beta$ product is additive.
q2. yes, that's right. But if you wanted to estimate $\alpha$ and $\beta$, Poisson regression wouldn't work because it log transforms the outcome. You'd have to fit a "custom" GLM having identity link function and a Poisson variance (so that the variance is equal to the mean as is the case in Poisson RVs). You might call such a model an "additive rate" model.
q3. The "additive risk model" is a GLM with binomial variance and identity link so that $E[Y|X] = \alpha + \beta X$ but $\text{var}(Y|X) = E[Y|X](1-E[Y|X])$ per the binomial variance assumption. You can also call it an "identity-binomial" model borrowing the terminology used elsewhere is calling the relative risk model a "log-binomial" model.
