In a logistic regression model, e.g.,
$$logit(\mu_i) = \beta_0 + \beta_1 X_{1i} + \beta_2X_{2i}$$
where $Y_i \sim Bernoulli(\mu_i)$, $expit(\beta_1)$ is the odds ratio.
In a poisson regression model with an offset, e.g.,
\begin{align*} log(\mu_i) &= log(\lambda_i t_i)\\ &= log(\lambda_i) + log(t_i)\\ &= \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + log(t_i) \end{align*}
where $Y_i \sim Poisson(\mu_i)$, $\lambda_i$ is the mean rate per unit time, and $t_i$ is some unit of time, then $exp(\beta_1)$ is the incidence rate ratio.
Now suppose that $t_i = 1$ for all $i = 1, \ldots, n$. So we have
$$log(\mu_i) = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i}$$
My question is, what is $exp(\beta_1)$ called? Is it still the incidence rate ratio (but with a unit time of 1) or is there another word for it?