iTunes automatically pauses when VLC plays a file
After a long search, I have realized that the setting to pause iTunes is not in iTunes, but it is on VLC. So it is not iTunes pausing the music because it detects another media player, but it is VLC which is pausing iTunes when playing files.

To disable this, just open the preferences of VLC and click on the following items:

Show All -> Interface -> Main interfaces -> macosx

After that, on the Control external music players field, you can select among Do nothing, Pause iTunes and Spotify and Pause and resume iTunes and Spotify. I selected the first option :-)

I share this knowledge here in Q&amp;A style in case someone else was struggling with this.