I started to track my finished routes and I cannot find anywhere a definiton of an attempt. Of course if I try to send the route, give up in falf, go down, have a snack and then go again and finish it then it's two attempts.
However what if I fall in middle, and after descending a bit I solve the part and send a route? Is it two attempts, too, or should I complete it again in one go? If so, is it then two or three attempts (does the first climb with fall count as one or two attempts)?
If you start a route, fall, lower down a bit, and then climb to the top, most people would not say that you have sent the route. See What is the term for completing a climbing route uncleanly?. Most people think of sending a route as climbing it cleanly.
If you start a route, fall, lower to the ground, and then climb the route cleanly, that is clearly 2 attempts. If you start a route, fall/hang and try the crux a 5 times, lower to the ground and then climb the route cleanly, that is not 2 attempts. At that point, it probably makes sense to just stop counting attempts.
An attempt begins when you leave the ground or unweight the rope and ends when you fall, reweight the rope, or clip the anchors. Thus, if you fall in middle and after descending a bit you make it to the top, you have taken two attempts. If you complete the route in one go on your next try, you will have sent the route using 3 attempts. That being said, a single number cannot fully describe the work put into redpointing a route. You can classify each attempt based on which section of the route you climbed during the attempt. For example, here is a nonexhaustive list of ways to classify your attempts:
Highpoint attempts: The number of attempts that began at the ground with the intent of climbing as high as possible before falling. NB: your highpoint is the highest point you have made it on any attempt which began on the ground.
Lowpoint attempts: The number of attempts beginning mid-route with the intent of climbing to the anchor. NB: your lowpoint is the lowest point you have started in a route and successfully made it to the anchor.
Crux attempts: For each crux sequence or move of a route, count attempts on the crux in the same way you would count attempts on a boulder problem. If a route has multiple crux sections, you can count attempts on each section separately.
In addition to counting attempts while projecting a route, your highpoint and lowpoint are useful metrics that show how close you are to sending the route. Once your lowpoint is lower than your highpoint, you should be able to one-hang the route.
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!
