Not really, especially if the roof is steep. Some customers do care about stuff like that.
If you mean flat side down, channel side up, then yes
Portrait means clamps on the long sides of the modules and landscape means clamps on the short sides. It seems simple enough to me.
That's news to me, I thought it was simply orientation when looking At something.
It depends on your frame of reference, but as long as I understand what they mean, I really don't care.
Ok Einstein
. Do we need to also know the radius of gyration of a module to determine if it's in portrait or landscape 
Portrait means taller than wide, like a picture of a person. Landscape means wider than tall, like a picture of land scenery.
Many (most?) module manufactures provide acceptable clamping zones on both the long sides and short sides of the frames, don't they?
On a groundmount the rails run vertical.
They do. The theoretical points that maximize strength are at a span of L/(1 + sqrt(2)/2), about L/1.7, where L is the module length. Compare this to what the manual specifies, and see how consistent it is.
I think what you are saying is that if you have a uniformly loaded beam supported at two points, the two support points that minimize the maximum bending moment within the beam are the points you describe. Is that correct? Or are you also considering the possibility that the beam is non-uniformly loaded, for any pattern not exceeding some uniform load?
Portrait vs landscape requires a notion of "up/down". From the perspective of the rails, it's not obvious what direction should be "up/down". Obviously you can make a choice (either parallel to the rails or perpendicular to the rails) and call that your standard. But why overload the terminology, why not use a different set of terms to refer to the rail frame of reference, e.g. long side clamping vs short side clamping?
From the perspective of the rails it doesn't make any difference. Call it whatever you like and so will the racking manufacturers. I know what they mean so it makes me no never mind.
My experience has always been that portrait and landscape refer to the perspective of a human standing upright and typically communicating to another human, who is also standing reasonably vertical. I don't see what the rails' perspective has to do with it, unless the supreme Court has expanded the citizens united decision and given rails personhood which sadly wouldn't surprise me.
Uniform loading is assumed. The cusps at the supports, have an equal and opposite bending moment as the peak at the center of the span. Here's the bending moment diagram:
So, to side with your position for a moment, if you are buying enphase Q cables, they have three different versions with different lengths Betwix the connectors, and they call them portrait and landscape. I admit that made sense to me immediately as the micros are mounted on the rail so its the relationship between the rails and the panels that matters not the absolute position of the panels. But I still say unless there is some specific context or use, the words should be used like you're looking at a picture or printing a document.
I don't have a "position"; I don't care one way or the other. I am just saying that I am not confused by their terminology. Clamps on the module frame long sides is portrait and clamps on the short sides is landscape, irrespective of the direction of the rails. They could be oriented at a 45 degree angle on the roof (I have done that) and it wouldn't make any difference.
OK, so if your beam member is only subjected to uniform loads, I get that the support points should be (sqrt(2)-1)/2 of the way in from each end, which agrees with your statement that the distance between supports is 1/1.71 of the beam length.
