Interesting....
I've come to the following conclusions:
Green Dot Speed = V_md (Clean Config)
Min Clean Speed = Min Manoeuvre Speed (Top of the yellow bar on the speed tape)
[Obviously a function of weight, CG location & MNVR MARGIN variable on the PERF FACTORS page - I believe the default value is 1.3g? Your company's assessment of the necessary safety margin can thus be ascertained by their view on what this setting should be.]
I concur with OSCAR YANKEE regards flap retraction schedule in the sense that I've never been told to fly anything other than the standard speeds which vary depending on whether you're below or above 53T.
I can't ever recall, having cleaned up, seeing a case where the top of the yellow bar is > 210kts? In this case (if indeed it is possible) the question that begs an answer is obvious...
But the "green dot speed" is a red herring and it would appear confusion about what it denotes is widespread, presumably because of what it is known as: "Flaps Up Manoeuveriung Speed".
To my mind, in the event of an ENG OUT, if terrain clearance is an issue, I will aim for "green dot speed" as it represents (L/D)_max for the clean configuration. This assures me of the minimum possible "glide" angle.
If on the other hand, range is an issue, I will fly the speeds the FMC spits out with the CI=0 i.e., Best Range Speed. Alternatively, one might choose to fly the LRC speeds - slightly less efficient. (I believe that having reached the ENG OUT MAX ALT at the ENG OUT driftdown TGT SPD, it may take half an hour to accelerate to LRC!)
If neither is an issue, perhaps because obvious diverts are numerous, the debate about the importance of what speed to fly at reduces proportionally.
Please correct me if I'm wrong.