Something stuck in my mind -- the extra hour from changing back the clocks

Obviously, we do not really gain or lose real time when we adjust the clocks.  Just our "official" time on the clock.  For some reason, this time around, I began wondering about that extra hour we "gain".

So, at 2 am, we turn it back to 1 am.  So on that date, in  this case November 5th, there are two hours considered 1 am to 2 am.  The first is usual, the second, manufactured when you set the clock back.  Obviously time still passes as usual.  But the thing that I am struck by, is that there are two instances of the same numerical time on the same day.

For instance, say a baby is born at 1:30 am.  Then you set the clock back again and another baby is born at the second 1:30 am of the day.  So both would have 1:30 am on 11/5 as their birth date, BUT they actually were born an hour apart.  

The same would be true for any "official" time  - of birth, of death, of a crime occurring... How do we distinguish the two 1:30s apart on paper -- on records.  Do we?  I don't think we do, but shouldn't we, since it does make a difference to when something actually occurred?

An interesting conundrum spinning through my mind.  Any thoughts?

Terri