Actual footage of the original version of Shinobi III is very difficult to come by, but here is what we've got so far:

This video was taken from a American SEGA video, showcasing some upcoming games. The Shinobi III segment isn't very long, but it does show part of the original's first round. The only enemies in this video are the ninjas and tigers. The tigers do not seem to come into contact with the ground that well, as some of them are seen walking around in the air. Also notice that there is no splash effect whenever anyone goes into the water.

This video was from a Spanish SEGA video. It is longer than the other video and shows a lot more stuff. The enemy ninjas are seen to be able to attack and leap around really fast, the samurai enemy goes into a defensive stance to block Joe's shuriken (something he doesn't do in the final game) and then of course we have the levels themselves -you get to see the airship level in action, among other stuff. This video also refers to the game as "Revenge of Shinobi II".

This video is also just as messy as the other one, with enemies frequently glitching through walls or popping up out of nowhere.

This video, uploaded by wadelyjp, shows more of the original game around 8 minutes and 40 seconds in. An early version of the track "Shinobi" can briefly be heard at the title screen. The video mainly shows gameplay from a level that looks similar to the prototype's seventh level, but is much more complete. However, the level design looks very uninspired and boring. The tiger enemies are still present, as is the ice ninjitsu magic. Later parts of the video show the sewer monster boss fight, who has a different projectile attack than he does in the final game.