I would go with Tecmo Super Baseball for the SNES.
While it doesn't have a MLB license, so no actual team names, I never noticed this as a kid. However, it does have a MLBPA license and includes digitized images of the players, which is pretty neat for 1994.
The game had a catchy rendition of 'Take Me Out to the Ball Game' as it's title theme and the crowd noises were what what would expect for a baseball game of the era. However, I'll always remember the 'Home Run' noise as the ball streaked out of the play field. Also, the use of the SNES's Mode 7 for fielding was pretty neat. I much prefer this to the overhead view of most baseball game where your players shrank to the size of ants. And the bullseye on the ground to locate where the ball was going to fall was also a nice touch.
Overall, I think it's the best baseball game on the SNES. Sure, Ken Griffey Jr.'s Winning Run looks and sounds absolutely amazing, but Tecmo Super Baseball came out right during the MLB strike of 1994 and it was a great way to still get your baseball fix.