The original scene for Eddie's introduction was a lot more vulgar than in the final game, being more gross and showing Eddie uncontrollably vomiting. It was wiped after the data for it was lost after a black out, and the scene was ultimately changed to be less over the top.

In a commercial for the game (I tried to find it but could not, sorry), you can hear Laura singing "ring around the rosie" before James comfronts her, which does not appear in the final game and may not even be canon.

However, the scenes you are talking about are only made for the intro of the game. Those scenes are not "deleted scenes", but more or less just scenes that were subtly thrown in the opening amv to give some backstory.