You mean the Quicktime events? In RE4, I found them to be unfair - they came out of nowhere, and I usually died the first time since I didn't realize that this was a cutscene that required input.

Heavy Rain, I haven't played, but from what I understand, it's similar (yet better in most key aspects) to the "predecessor in spirit" Fahrenheit/Indigo Prophecy, right?
This is actually a real interesting point you bring up. It boils down to the one question, that is: How do we define a game? And if this is done via game mechanics, the follow-up question must be: Is that still a fair assessment? Because as I've said above, I believe that so many things come together beyond the game mechanics, that this approach wouldn't do this complex medium justice.

Heavy Rain, as well as Fahrenheit, seem to be mostly about decisions. Disregarding the quick time events, that I can't say anything about since I haven't played the game, it seems to me that the game is content in providing you an experience. A story - putting you in an extraordinary situation, asking you how to deal with it. You're tasked with finding out how you react in situations, whom and how many people (and what kind of people!) you trust. In that sense, however, Heavy Rain and Fahrenheit are radically different - their stories are different, their setting is, their characters are. The fact that they feel so different, then, is supporting the idea that it is not about mere mechanics (those being mostly a dialogue-system and a way to move your character to certain points that, again, represent decisions), but that the entire experience is shaped by so much more.

Here's an observation I'll put out here. These kind of games - and maybe Minecraft as well, they seem to represent a different "state" of gaming.
Compare (and contrast?) to how you played games as a child. It was about make-believing, about putting you in situations. Creating things with LEGO was not necessarily fun because you enjoyed putting those bricks together, or even because you liked the end result. The process itself got it's meaning mostly from the vivid imagination that most of us seem to lose at a certain age.
Later on, as adults, we play different kind of games. We may still enjoy a board game with a setting (Check this out!), but the "classic" games, say poker, or most other card games, they are about their mechanics. Chess, too. They represent a struggle, sure, but the reason we play them is not to find out what heroes or leaders may have felt on the battlefield. We play them because we find the tactics, the psychology behind it intriguing.
This seems similar to the definition usually brought up about video games. At it's core sits the game mechanic, and that rules all.
But we've seen that it can be difficult to accuralty describe (and differentiate - just as important) games. We fail to see the appeal in Minecraft, if there doesn't seem to be a clear end goal in sight (see spikes post). However, thinking back to the earlier kind of play more common in how we used to approach games as children, things seem to be similar. Is, then, Minecraft just a tool to play games just like we used to? What we lost in "vividness" of imagination provided by Minecraft's sometimes crude graphics, being the spark that sets your imaginary world on fire?

Different people prefer different games. Minecraft and Terraria and those all invite you to make your own story up as you go along - possibly even without you consciously doing this.