Initially, I had only heard of MLE and use it for almost everything, e.g. point estimate and model selection (with some penalty).
Then, MSE appeared, which seems to play the same role as MLE does. I learned that both MSE and MLE are results from loss functions (entropy vs quadratic). Choose which one to use is completely up to one's belief.
Now, I came across stuff called scoring rule. There are new terms like Logarithmic scoring function and Brier scoring function. However, I feel they are just another names of MLE and MSE methodology. Minimize Brier is essentially minimizing MSE, and Maximize Logarithmic is exactly the same as MLE.
Am I right here? Why do we have different names for the exact same things between scoring rule and loss function?