Since we are using the logistic function to transform a linear combination of the input into a non-linear output, how can logistic regression be considered a linear classifier?

Linear regression is just like a neural network without the hidden layer, so why are neural networks considered non-linear classifiers and logistic regression is linear?

Logistic regression is linear in the sense that the predictions can be written as
$$ \hat{p} = \frac{1}{1 + e^{-\hat{\mu}}}, \text{ where } \hat{\mu} = \hat{\theta} \cdot x. $$
Thus, the prediction can be written in terms of $\hat{\mu}$, which is a linear function of $x$. (More precisely, the predicted log-odds is a linear function of $x$.)

Conversely, there is no way to summarize the output of a neural network in terms of a linear function of $x$, and that is why neural networks are called non-linear.

Also, for logistic regression, the decision boundary $\{x:\hat{p} = 0.5\}$ is linear: it's the solution to $\hat{\theta} \cdot x = 0$. The decision boundary of a neural network is in general not linear.
