Friends,
Let me make it a bit easy for you guys, the question is like this:
If Superman, who was born on planet Krypton -- a planet that blew up shortly after he escaped -- is being asked where he was born, should he truthfully state Krypton or make a false statement such as Kitchener or Kampala?
You were born where you were born, put the name of the country as it was when you were born i.e. as in BC/PP etc. It would be wrong to change, as the concept of many new nations didn't exist at that time.
If you keep changing Leningrad will be St. Petersburg too. As I said earlier, to finally end up having many mismatching documents to face unintended consequences. You can put it as Leningrad, USSR while your kids can say St. Petersburg, Russia. Nothing wrong in it.
If you were a sports lover and watch a lot of matches, you will know this in a flash second. When you look up a player, listed is the country they were born in, not what it is today.
Here is an example
http://sports.yahoo.com/nhl/players/1851
Same player, another site
http://espn.go.com/nhl/player/_/id/358/jochen-hecht
Another one
http://sports.yahoo.com/nhl/players/3233
One more example (different country) Now Kazakhstan
http://sports.yahoo.com/nhl/players/1884