maximize likelihood = minimize K-L divergence = minimize cross entropy with "maximize likelihood" being the preferred term (in most disciplines except maybe yours).

LogitModelFit and GeneralizedLinearModelFit can give the exact same estimates:

data = {{1, 0}, {2, 0}, {2, 0}, {2, 1}, {2, 0}, {3, 0}, {3, 0}, {3, 0}, {3, 1}, {3, 1}, {3, 1}, 
  {4, 1}, {4, 1}, {5, 0}, {6, 1}, {7, 1}};
LogitModelFit[data, x, x]["ParameterTable"]

GeneralizedLinearModelFit[data, x, x, ExponentialFamily -> "Binomial"]["ParameterTable"]

I don't understand why it concerns you that Mathematica "treats a logistic model as a special case of a generalized linear model."

LogitModelFit and GeneralizedLinearModelFit use maximum likelihood to estimate the parameters. R's glm and most generalized linear model software also use maximum likelihood.

But minimizing K-L divergence and minimizing cross entropy are equivalent to maximum likelihood. You must be from a machine language background rather than a statistics background. Yes, another case where a relatively new discipline thinks they need to create a whole new language.