Perceptron learning rate irrelevant in convergence
Your conclusion is correct. 

Note that the classifier is of the form of 

$$f(x; \theta) = \operatorname{sign}(\theta^Tx)$$

while the update rule is 

$$\theta^{(k+1)}=\theta^{(k)}+\eta y_tx_t$$ which only occurs when there is a misclassification and we only care about the in the classification but 

$$\operatorname{sign}(\theta^Tx)=\operatorname{sign}(\eta \theta^Tx)$$

as long as $\eta$ is positive.

It only scales $\theta^Tx$ but our concern is just the sign, hence the learning rate is irrelevant.

Remark: You might like to note the assumptions being used in the proof though. For instance, there is this assumption that $\theta^*$ exists. It is assumed that the data can be perfectly separated by a hyperplane.