While in a boring meeting at work, today, here's what I think I figured out about the ambient display...
I
think the ambient display won't come until my phone has locked itself. i.e., I have my phone set so that it won't lock until 2 minutes after I turn off the screen. So, if I turn off the screen and put it down on the table, I'm unable to get the ambient display to kick in until two minutes have elapsed and the phone becomes locked. (If that's true, then I have no idea what that means to an phone that uses no lockscreen.)
To get the ambient display to kick in, I have to lift the phone to a nearly 90 degree angle from its resting position. (Maybe 75, maybe 80; sorry I didn't have protractor with me at this meeting

) Nudging it doesn't do a thing. Lifting it up partially to, say, a 30 degree angle won't do a thing. Gotta just grab it and lift it to be perpendicular to what its resting position was. When I repeatedly did that, I could get the ambient display to reliably come on every time. Didn't matter if I was lifting it sideways (so I was holding it in landscape, or if I lifted vertically (so I was holding it in portrait) - either way worked.