Actually your right about the running back used as a generic Term for a category, such as lineback instead of left outside lineback, but Half backs are used in West Coast or Vertical philophys, such as the Half back in Pro set forms are used as Tight-Ends and in I-Formations they are used as Fullbacks, but in Other Offense philophys they are used as Fullbacks.
Yes, a half back is basically the same thing as a running back. A Half Back is an old way to describe a player in the backfield in football. There are three distinct positions that first arose in footbal for players behind the line of scrimmage. The quarterback, the half back and the full back. The quarterback was the closest to the line of scrimmage, the full back was the furthest fom the LOS and the half back was in the middle, you guessed half way between the two.
Technically there is no difference. The term "Running Back" is generally used to describe Fullbacks, Tail Backs and Half Backs. - As it used to be, full backs were the faster, smaller players in the backfield, and halfbacks were the bigger, stronger blockers. But, as the positions evolved, half backs/ tail backs are now the "feature" backs. Typically smaller and faster, and get more carries than today's full back, which is now the bigger, stronger player, who typically block for half/tail backs. - Running Backs include tail/half backs. Such as Ladanian Tomlinson, Barry Sanders, Gale Sayers, Willie Parker, etc. - Full Backs are the bigger stronger backs, and are considered a separate category from the "other" running backs. Full backs today are players like Mike Alstott, Lorenzo Neal, etc So to be explicit, when someone thinks of a "running back" they're typically thinking of a tail back, or half back, as it's more commonly referred to.