Hopefully someone can clear this up, since I couldn't find a complete answer.

Which is the best version of this game?
As far as I remember, there are two versions: The original and the one with "simplified" puzzles/animations but improved graphics.

Is this correct? And are there any other versions? How can I tell which version is which (by the packaging?).
As I understand it the gog-version is the "simplified" version?

Amerzone for Windows is in normal version, Amerzone for Mac OS X is simplified crap with better movies. Gog has them both :) Gog unfortunately does not provide the normal version for Macs so you must play via emulators, virtual machines or Bootcamp.

Thanks.
Is there a game version with the better movies/graphics, but NOT simplified at the same time (physical or online)?

I know that they do a iOS version that look very similar to Mac version. Here you have some press inforamtions: http://gamasutra.com/view/pressreleases/198939/Remastered_Full_version_and_new_Freemium_of_Amerzone_TheExplorerrsquos_Legacy_is_available_on_iOS.php. If Mac and iOS versions are the same and if they do the same version for PC then, there is no sense to worry about better version (than standard gog) because the playability was more important than graphic. Maybe also read this topic about DVD verson: http://www.gog.com/forum/amerzone_the_explorer_s_legacy/graphics_quality. Maybe author of this topic can send you some screenshots from DVD and normal edition.
The game is very short and probably you spend more time looking for answers than playing it :). If you still want to look for answers you can send a email to the game developers - Microids company and maybe they will help you.

There was a DVD-version released around the time of the original. It has much better looking videos, but unfortunately I've never got them to play in-game (need hardware Mpeg support I think).

There is also a PS1 version I saw play on youtube but I don't know exactly how it differs (seemed to have blurrier graphics and load scenes slower, also had slightly different cursor UI, but i'm sure there are other differences).

I am unsure of the differences, but a list of releases is here:

https://www.mobygames.com/game/3973/amerzone-the-explorers-legacy/