Since $Y_i$ is a binary variable, its distribution is the Bernoulli distribution:
$$Y_i | \mathbf{x}, \mathbf{w} \sim \text{Bern} \Bigg( \text{Prob} = \frac{\exp(\mathbf{w}^\text{T} \mathbf{x})}{1 + \exp(\mathbf{w}^\text{T} \mathbf{x})} \Bigg).$$
One alternative way of looking at the logistic regression is to regard the observed response variable as a discretisation of an underlying "latent variable", where the latter has a logistic distribution. In this (equivalent) alternative formulation, we have an observed response variable $Y_i \equiv \mathbb{I}(\tilde{Y}_i > 0)$, with the underlying latent response having the distribution:
$$\tilde{Y}_i | \mathbf{x}, \mathbf{w} \sim \text{Logistic} \Bigg( \text{Location} = \mathbf{w}^\text{T} \mathbf{x}, \ \text{Scale} = 1 \Bigg).$$