It appears that some of these tunes do not encode properly with most of the players, they sound differently.
That's because module playback is not standardized at all, and most tracker developers documented their file formates scarcely, if at all. The correct interpretation of effects is often guesswork and hard to get correct without reverse-engineering or proper documentation.
This is how it's been reproduced by the players:
This is actually the correct version and how the tune is supposed to sound. Read below why that is the case.
This is how it's supposed to be like:
Since you tagged the file with "UMX", I guess that this is the version how you recorded it from Unreal, Unreal Tournament, Deus Ex or a similar game. And that gets us straight to the point: The Unreal sound engine (aka Galaxy Sound System) was designed to handle XM modules. The file you are trying to play is an IT file, which has many advanced features that are not supported by XM. Galaxy can play IT files, but it does not support most IT features. In this particular case, it doesn't support "New Note Actions", which are used to keep a note playing while a new note is triggered on the same channel. Galaxy simply cuts notes and does not continue to play them, that's why you get the "ragged" sound at the beginning of your recording. So the version you described as "this is how it's supposed to be like" is actually the
wrong version because Galaxy is simply not capable of playing the module. I'm not even sure how on earth one could think that the crap Galaxy makes out of this module could be the "correct" version. It's totally broken and horrible.
What player should I use to get the real reproduction of the tune(s)?
The tracker that was used to create the tune, of course (in this case ModPlug Tracker / OpenMPT).