No, a bit more complicated and weird than that. Great guess though.
So the years 2002 and 2003 were "trial runs" for moving from 11 games to 12 games allowed in the regular season. But at that time they also allowed teams to play an extra "kickoff game" in what we now call Week 0 (weekend before Labor). So teams were technically allowed to play 13 regular season games.
Why those two years? Because they happened to be years where Labor day fell on a Sept 1 or 2, which means Aug 30 or 31 is like an "extra" weekend to the regular season, in a sense.
Assuming they looked at how many players got injured per game, didn't see a drastic increase in rate going from 11 to 12, and called it a day. We've had 12 per year since, regardless if we have that "extra" weekend or not.
I don't know when playing a Kickoff game was no longer exempt, but pretty sure it no longer is. You can still do them of course, but you will "lose" a home game to do that. (was probably what they were trying to avoid, back then)
Week 0 scheduling is allowed now for any team that plays Hawaii at some point in the season (and possibly some other exemptions), and of course any team they schedule during that week.