If the cache wasn't there then it would be a DNF. You certainly wouldn't log it as found if it wasn't there. Doesn't seem there are too many other options.
Cache there and found means a found it log. Cache not found (regardless of why) means a DNF log. Simple.
Oh, and DNF's aren't logged enough as it is so I wouldn't vote to add the stat to public profiles.
I disagree a little. When I can't find a cache, it usually means I'm looking too hard or looking in the wrong place ("Didn't find it" or "DNF"). By default I assume error on my part. However, if I have reasonable suspicion that it's truly missing I file a "Needs Maintenance". "Needs Maintenance" also applies when the box has been destroyed or its the box is found, but contents are missing (e.g. multi- or puzzle-caches).
Perhaps I'm too naive, but I expected going into Geocaching that the caches would be more disguised. I look so hard for the hollowed out rusty bolts, fake sprinkler heads and planted tree stumps that when a real cache is missing, I keep assuming I missed it. It's only if another Geocacher who has found the cache in the past speaks up that I realize the cache box was actually stolen, though I start to suspect if two other cachers surrounding my DNF log post also post a DNF.
For the record, I don't post each of my attempts. For example, there are two caches... both in the perimeter of children's parks that I keep frequenting. If I don't have any new information on my DNF, I don't repost after the first one. If, however, I discover something that could be helpful to the cacher who maintains it or to following cachers looking for it, I'll post another DNF with additional info. I had actually searched for the second stage's cache for "Wye ask Y" five times before posting a "Needs Maintenance", but only posted 1 DNF.
Someone mentioned they didn't want to have their DNFs publicized because they don't want to be judged. I'd have to say to those who think this way: Showing your finds without your did-not-finds is wrongful representation and you are therefore advocating deceitful practices. I have big trust issues with you.
There are other valid reasons not to show the DNFs in a profile, but there are even more valid reasons not to show any finds to the general public in a profile.