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"]

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

You're right, empirically! I tested the LogitModelFit against a manual minimization of a cross entropy and they give the same optimal parameters. I was concerned because I didn't know what statistical assumptions are used in the LogitModelFit for the independent trials of generating the dependent variable, Bernoulli or Gaussian? I guess Bernoulli, otherwise maximum likelihood would give a chi square type loss function instead of the correct cross entropy.