Confusion matrix, when mistakes below diagonal are better than above the diagonal
You are incorrect when you say accuracy and precision assume the same mistakes. They are quite different in nature as to when they are applied for separate use-cases.
For more details on how they are different - https://towardsdatascience.com/accuracy-precision-recall-or-f1-331fb37c5cb9
As per your situation, you could opt for precision as you are looking only for the right answers, in case of a spam email detection.
There is no metric as such which calculates how much your model underfits or overfits but a genera idea can be made when you compare the validation curve and the training curve.