Finally after a few weeks, I have managed to solve this particular issue. It turned out to be a simple solution, as simple solutions are most often the hardest ones to find...
It turns out that Embarcadero RAD Studio has the Borland.Data.DbxCommonDriver.dll, Borland.Data.AdoDBXClient.dll and the Borland.Data.DBXInterBaseDriver.dll installed locally as well. These dll's have a different version number than the DLL's provided by the ADO.NET 2.0 drivers provided by Embarcadero. I tracked this down by performing a search in the Windows Registry. So if you are facing an equal problem, find out what drivers are registered in the registry to find their exact location...
After removing the references that pointed towards the ADO.NET 2.0 drivers (typically installed at C:\Embarcadero\Interbase_ADO.NET) and setting the references to the DLL's located at the Embarcadero RAD Studio installation folder, things were still working locally.
After publishing the site, it still did not work. So I removed every reference from the machine.config files and updated the web.config file for the webapplication. Finally, I deployed the files 'dbxconnections.ini' and 'dbxdrivers.ini' alongside my webapplication (which was stated in this Embarcadero knowledgebase article).
After cleaning the server, I did another publish of the webapplication and it works! It still is a pity however that there are so many different versions available of these Borland.Data.DbxXXX drivers, things might get tricky to really find out what is wrong.
Though I still doubt that there are not many programmers facing this particular issue, I still thought it might be handy to post my solution. I do hope it may help :)