I often read that: If we believe that the functional form of the dependent variable is a cumulative normal density, we may use probit and if we believe that the dependent variable follows a logistic response. In this case, we should use logistic regression. How can I deduce this in practice
If I'm understanding this correctly, you're essentially just asking how to assess whether or not a given statistical model accurately fits a set of observations, with specific questions relating to differing between probit vs. logit error distributions. Unless you know the assumptions behind how the data was sampled, you cannot "deduce" the distribution other than really applying goodness of fit tests to get an understanding of how well the discrepancies match the models.
These are two uniquely separate and distinct distributions with different assumptions embedded in their derivation so aren't necessarily the only choices for distributions of binary categorical errors, however they do tend to be tested together as they both have nice properties and supports for glms where the distributions of the errors are binary categorical data. And unless you know specifically the data was sampled from the distributions like you have stated above i.e. probit from a cumulative normal or logit from a logistic response, you cannot really "deduce" this in practice in any better way then these goodness of fit tests and comparisons.