With the deadlift (and other lifts that begin with the bar at a dead stop on the floor, such as the Olympic lifts), then taking a second or two between reps is usual, and arguably safer than doing the reps touch-and-go style (where you pretty much "bounce" the bar off the floor).
The main reason, again, pertaining to deadlifts mainly, is to allow you to reset your body and pull with decent form (i.e. not rounding your lumbar spin overly much).
With other lifts where you support the bar (i.e. bench, squat, overhead press), my opinion has always been that if you're supporting the weight of the bar, then you're still working in the same set, because your body is still working. During high rep sets of squats, I've stood gasping for air for several seconds before doing another rep. During the bench press I consider it normal to take a breath between reps with the bar at the top, which can take a second or two, with the overhead press, I rest the bar at my shoulders for a second to kill any momentum before I press again.
The best way to think about it is to be honest with yourself and not worry about it too much. If you walk away from the bar, then the set is over (I used to say if you let go of the bar, but I've seen some powerlifters stand up to take a deep breath between reps), if, in your head, you're still in deadlift mode, but need to shake our briefly, then I'd say it's the same set.