There are some cases where the finer details of what constitutes an "attempt" matters.
First, completing a climb as an
on-sight (on first attempt, without any prior knowledge) or a
flash (on first attempt, but with some degree of prior knowledge ot "beta") is considered better than completing it after several tries. This gives the climber more "bragging rights" and a higher ranking in any (more or less formal) ranking lists.
While most would treat any clean ascent (
redpoint) not on-sight or flash as equal, some like to point out that they "sent" the route on the second or third try. (As opposed to spending weeks or more working on it) Also, in bouldering competetitions, a climber usually has three attempts on each route (or "problem")
How strict the rules for when an attempt starts and ends depends on the context. In competitive sports climbing rulesa are naturally more strict. (There is usually a time limit as well as no weighting the rope). Outside competitions there is more of a gentlemans agreement, where an attempt starts as soon as you touch the rock (or an artificial hold) and ends when the rope is weighted.
Note that downclimbing to a good rest position or even to the ground might be allowed!