Yes, it does seem that touching the phone quickly has something to do with it. Meaning, when I unlock the iPhone (using Touch ID) and then IMMEDIATELY touch one of the app icons. I've never had it respring when I'm in the middle of already using the phone. Only when I first unlock it.
I visited the Genius Bar today and the Apple Genius pulled the diagnostics from my phone onto his iPad. He found evidence of a lot of different apps crashing and said it definitely wasn't a hardware issue (no Panic.plist files). Although he was somewhat evasive about how often he sees complaints about this respringing issue, he did comment about the fact that they've seen some repeat bugs popping up with various versions of iOS 7. He said Apple will quash a bug in, say, 7.0.2, it will remain quashed in 7.0.3, and then the bug rears its ugly head again in 7.0.4. He said that 7.1 is in beta (which I already knew) and he believed it would take care of a lot of the issues users are seeing.
At this point, I'm just going to leave Reduce Motion turned off. The phone hasn't respringed all day long, despite my attempts to make it do so. Hopefully, that will get me to 7.1 without too much hassle.
Mark