The futuristic city with flying cars at the beginning of the movie definitely has all the cyberpunk visuals, and technically Zorg is the head of an evil corporation. But the real villain of the movie is incoming force of evil/darkness. And the plot is resolved through the power of love. Even the President of Earth is actively trying to help do the right thing and save the planet.
While there are some great cyberpunk visuals at the beginning of the movie, I don't know if the themes are there to call this cyberpunk. What do you think? Would you consider The Fifth Element to be cyberpunk?
Here is a good cut list comparing the 1981 version and the more recent version.
Consider some other cyber punk classics that are contemporaneous with the older version: Akira, Ghost in the Shell, Robocop, and Bladerunner. The older version hits many of the same beats story wise, especially with Bladerunner and Robocop. They have that very crime-drama/ film noir styleing, that the source material very much so has. Also, the original I think does show System Thats Grinding People Down, and the corrupt/ defanged system by corrupt cops. For example, Corban has to bribe a cop to get any services in the original, is turned into a cop refusing to act until he gets his McDonalds served to him in the more modern version.
I personally consider the original to be a cyberpunk classic that actually helped form/ shape the genre, but like many genre defining pieces, it doesn't actually perfectly fit what the genre becomes. I mean you figure bladerunner comes out almost two years after the original fifth element comes out as a short in the Heavy Metal movie.
I didn't realize it was a remake of a Heavy Metal short. Thanks for the link!