Bmax, the logbook has a few purposes. One is, you can prove your experience to interested parties, such as Instructors and Dive Boat Captains. Secondly, you can see your own progression and training development. Thirdly, if you are diving more than once on the same day, you use the logged dive to plan the next dive and figure out your residual nitrogen and maximum dive time. Finally, you can return to your logbook for details if you dive the same site again, eg weighting, temp, current, what suit you used, etc. Some divers draw a sketch of a wreck if that was what the objective was, identify fish they saw, etc etc. Good to have the logbook entry signed by your buddy or Dive Guide or Instructor or Dive Operator so it is more verifiable.
Instructors have to follow guidelines as to what constitutes a dive so as to maintain quality educational standards. In PADI, an open water dive is at least 15 feet deep, and either 50 cubic feet of gas breathed, or 20 minutes bottom time. NAUI is similar. Don't know about SSI but it is probably similar. A dive is breathing compressed gas under water and can be very short or long. For example, if you had to abort the dive for any reason, that is an interesting dive. How did you handle it? What was the reason?
So this dive would be logged and has nothing to do with the educational minimum standards. As was said, it's up to you to log whatever you want to log. More is better than less.
There should be a minimum of at least 10 minutes surface interval between dives, for the purposes of dive tables, otherwise it is all one dive. So if you come up during the dive to the surface, and go down again, it counts it as one dive, both for log, and the tables.
The main thing is, get a log book and get in the habit of recording all the dive's information, as soon as you get out of the water. Bring the log book to all your dives, keep it in your dry, not gear, bag. Also some people transfer log book data from their computer so they can better analyse their dive profile and what happened.