From a stats perspective the cutoff is included in the coefficients. If you use a design matrix (add a column of 1s to your variables) you get in a non matrix notation (beta_01 + beta_1 X_1 +...) So the threshold can be considered beta_0.

In the software, you can get classification models to output class probabilities instead of class labels. You can then use whatever threshold you like for to transform those probabilities to labels.

You may see it refered to as "discrimination threshold". Varying that threshold is how ROC curves are constructed.