In linear regression, I train the model so the graph runs best through the data points, so the geometric distance between f(x) and $y^i$ is minimized.
Now, is it correct that in logistic regression I do not try to fit the graph perfectly through the data but instead try to mimimize the error in the predicted probabilities? Why is that a difference? And what happens when I plot the resulting graph to the dataset. Does it still look like it's perfectly fitted through the data, and why?