(i.e. > 0.5 equals 1, and < 0.5 equals 0).
This seems to be a "rarely-questioned assumption".
Is there a reason why this is considered reasonable? And is there a name for the cut-off (i.e., if I were to want to change the cut-off, what keyword should I search for inside the software's manual?)?
Almost all statistical models give you probabilities and it is up to the domain to determine the cutoff. You can clearly see this in logistic regression it's giving you a probability not a 1 or 0.
When a clinician gives me a dataset I build the model and leave it to them to do the cut off. It is not my domain or expertise to tell them where the cut off is. Plus it gives away the responsibility. I'm not responsible for it unless they don't know and are asking me for a reasonable cut off.
An example is people opting for artificial in semenation because of problem. I believe the cut off is above 60% because of the trade off.
You can read more about it in Regression Modeling Strategies by Dr. Frank Harrell.
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.