Dave
The terms I have encountered in reference to a man with an unauthorized absence from his unit during the Civil War are "absent without leave -- AWOL in modern parlance, and deserter. The distinction between the two in the modern American military Uniform Code of Military Justice is that a person declared AWOL was considered to have left the unit with some indication or judgment being made that the soldier intended to return. On the other hand, a man was declared a deserter either because his command had some evidence that the soldier did not intend to return, or a certain amount of time had elapsed following his absence. I think in modern times a man was declared a deserter after a 30-day unauthorized absence.

As best as I can tell, the judgment in the Civil War with regard to the distinction between AWOL and deserter was not as tightly drawn as it is in the modern miitary. However, commanders used some criteria to make a distinction between the two. I think it mostly had to do with the commander's judgement as to whether a soldier intended to return. If you look at court-martial records you can see where a man is charged with desertion, but is actually found guilty of the lesser offense of AWOL. So the court members clearly had some criteria in mind to make the distinction. In Jack Bunch's "Military Justice in the Confederate States Armies," Bunch notes that the Articles of War did not define either absence without leave or desertion but Articles 21, 41, and 50 refer to various forms of unauthorized absence. According to Bunch the distinction between the two terms is one of intent. If the court heard evidence that the soldier returned of his own volition to the unit, or the evidence showed he intended to return, then the court was likely to find the soldier only guilty of AWOL. Howver, if the court found that the accused did not intend to return, then it would find the soldier guilty of desertion