What is considered an error in the sustained attention to response task (SART)?
Both are errors indeed. Pressing when a 3 is shown is called a false alarm, and not pressing when a three is shown is a miss. Correct responses are (1) hits (pressing when anything but a three is shown) or (2) correct rejections (not pressing with a three). 



These classifications are used in signal detection theory (Wiki). More specifically, they are used to calculate sensitivity and bias.

Sensitivity, or d-prime (see also other questions on CogSci, e.g. Calculating d-prime), is the metric for how well one can distinguish targets from distractors. Bias, on the other hand, is:


  the extent to which one response is more probable than another (Wiki)