There is seemingly a difference whether one approaches the "logit" from statistics / econometrics:
$$F^{-1}(\pi) = \log\left(\frac{\pi}{1-\pi}\right) = \text{logit}(\pi)$$ (I)
Or from the machine learning perspective, where it is typically called "logistic regression":
$$F(\eta) = \frac{\exp(\eta)}{1 + \exp(\eta)} = \frac{1}{1 + \exp(-\eta)}$$ (II)
There is essentially no difference between what is done to classify new observations. But what exactly does that mean the inverse of the CDF? Is there a possibility to reformulate (I) to get to (II)?
Thanks!