Statisticians do not fit models by hand, they "fit" models in the same way machine learning pratictioners "train" models. "Fitting" in statistics lingo is completely synonymous with "training" in ML lingo. Statisticians were the original ML practitioners.
I will second this. I use the two words interchangeably. I would probably use the word fit when talking about liner regression and train when talking about a more complex ML model that requires more computation. I really don't remember the last time I fitted a regression model by hand, it would have probably been in a college exam.
