Not necessarily, but it has been a while since I was in the Army. Sometimes a lower-ranked person in a class of ranks will be referred to by the class name -- so you might call an E-1 a 'Private' out of courtesy. Similarly, you'll see 2nd Lieutenants (2LT) referred to simply as 'Lieutenant' and even Lieutenant Colonels (LTC) referred to as 'Colonel'.

Looking at the DOD rank page, it seems as though the term 'Recruit' has been dropped for the Army (although it is still there for the Navy, and an E-1 is a 'Private' while an E-2 is a 'Private E-2'. I guess I have to move with the times. :)