I have never gotten what the meaning of the adventure genre is? Is it like a light-novel type of game, since that is what heavy rain represents and I have also seen Ace Attorney and Layton within that genre.
You're pretty much correct on your assumption. Point and Click adventures like The Curse of Monkey Island and Grim Fandango are the superstars of the Adventure genre (despite not making it on this list...), but newer games like Phoenix Wright and Professor Layton (this one is an Adventure-Puzzle hybrid) have lead to a resnaissance in the genre.
Talk to Okr if you have any other questions about the Adventure genre. He is the master.