When I got my first —original— Xbox in 2002 I bought it what already was a modified, pirated version of the console. It was ready to store game copies onto its hard disk and let players play them easily and with only one disadvantage (to the user): you could get banned from the Xbox Live service that was just starting to show its potential in those days.
When the original Xbox was launched, the PS2 had been already available for a year and a half, and that head start was a big obstacle for the Xbox, which nonetheless was able to gain traction and become a worthy alternative. The PS2 dominated that sixth generation of consoles, though: 150 million units sold as of February 2011, while the Microsoft Xbox had sold over 24 million units as of May 2006.
Both of them suffered greatly from piracy, and one could wonder if the impact of that piracy was bad or good for the original Xbox. The competition against the PS2 was really tough, but having access to an easy way to modify and take advantage of game copies downloaded from the internet may have had a positive impact for users that were debating which one of the two big rivals get.
The ability to run those copies from the hard disk and projects such as XBMC were specially interesting for certain users that understood that the console could be a great alternative as a Media Center. I used those capabilities too, and I can confirm that as far as I can recall the original Xbox was much more interesting than the PS2 for me.