I don't know how to define a logged dive, but like pornography, I'll know it when I sea it. (Like what I did there?)
But seriously, I tend to log anything in the open water, even if it is short if I learned something. And short dives pretty much always have some interesting reason for being short. Case in point: Dive 200.
Dive 200 was a night dive on Osprey Reef, some 230 km off the coast of Australia. The late afternoon dive had been a shark attraction dive, so there were several reef sharks in the water when my husband and I and couple hilarious Aussie guys splashed. Also, it was raining so there was no moonlight. The water was like driving in a snowstorm with loads of organic particles and we never managed the not-completely-straightforward navigation to the reef. Instead, we got pushed WAY off the the dive site in the current during the time it took us to realize we weren't going to find the site and do a safety stop. Okay, that safety stop wasn't really necessary, but we did one to be, well, safe. It was a liveaboard and we were logging lots of bottom time. We surfaced after maybe 12 or 15 minutes (I'd have to go back to check--and I can because I logged it), and roughly a quarter to a third of a mile from the boat. We signaled the boat, they signaled back, and they already had the zodiac in the water to pick us up. It's the only time I've had to be hauled out of the ocean. When I got onboard, I joked about how I, as the only woman was the last one "rescued" and how that was a memorable 200th dive. The DM tried to convince me I didn't have to count it because it was less than 20 minutes, but my reaction was "hell, no. I'm counting it." I learned some things and it was super memorable--carried out to sea on a stormy night in shark-infested waters.