It is not two phase power with two legs 180 degrees out of phase.
Well, yes it is. To quote from the summary at http://www.allaboutcircuits.com/vol_2/chpt_10/2.html (at which you ought to have a look)
"A split-phase power system is one where there are two voltage sources, 180o phase-shifted from each other, powering a two series-connected loads. The advantage of this is the ability to have lower conductor currents while maintaining low load voltages for safety reasons."
You can argue that this is something (in fact the first thing) I found on the web searching for 'polyphase neutral' but I have never seen anything published nor, up until this point in my life, met anyone that contradicts what it says.
That connection of the o-scope doesn't hold water for this purpose. Neutral isn't the correct reference in this regard. Use the waveform math on that scope and add those two together. Do you get 240V or zero?
I get 240 because I add them in the proper phase as happens when the transformer coils are connected correctly. One channel is connected from L2 to neutral and the other is connected from L1 to neutral so they are 180 ° out of phase. L2 is thus negative with respect to the neutral when L1 is positive WRT it but that means that the neutral is positive with respect to L1. Thus we are connected - + - + just like the batteries I mentioned in an earlier post and we get summation of the voltages. Take your transformer drawing (upper one) put in a load and do Kirchoff's law around the loop. Bottom to neutral is plus 120 V (bottom is negative with respect to neutral). Neutral to top is 120 V. They are connected in series positive to negative. The voltages add.
But at least I now know what the source of your confusion is. You don't understand that the reference is always taken at the neutral point in a Y connected system and perhaps don't understand that a biphase system is a degenerate Y connected 3 phase system.
You have to use an end point for reference because it's the only place you can get 120 AND 240, and as you see below they are in phase and that the voltage on CH B is 2x that of CH A because it's taken at the correct reference for a 120/240V circuit (disclaimer, I'm using a small step down transformer from the line but net result is the same, just lower voltage).
Not so. Using the center as the reference my scope picture clearly shows two 120 volt circuits. If I connect them in series I get 240V across the outside.
But the essence of your problem is the reference point and as I noted earlier this is arbitrary and mentioned that in a Y connected system the neutral is the only sensible reference (which is why it is used as such). So lets suppose we are in an industrial or office building fed with 3 phase Y connected power as they often are. It is common to have both 120V and 208 V circuits (for lights, floor polishers etc.). A given panel has two (of the three) phases connected to its breaker bars and the neutral connected to the neutral bar. It's a biphase system with the phases 120° apart instead of 180°. If we did my experiment or yours (I did notice that you changed the photograph from one with 2 different references to one with a common reference so I'm assuming the later is the one you want to convey) we would get a picture like the one below. I would get the green and blue traces. You would get one or the other and the black trace. If I moved the reference line on one of my channels to the the opposite phase I would get the black trace as well. Note that 100 here on the voltage scale represents 100% of either rms or peak voltage.
Now in this picture with the references connected my way the two phases are not 180° apart and connected your way the one phase and the sum are not in phase. How would you argue here? You couldn't say this is not a polyphase system and that the two phases not 120° apart because they clearly are. You cannot say the 120V and 240V circuits are in phase because they clearly aren't. The only difference between this 120° arrangement and the 180° one is the phase angle is 180 ° and that happens to be the phase angle that brings the phase to neutral voltage into alignment with the phase to phase voltage.
Now in a delta connected system there is no neutral and you have to pick a phase or phases (in a 2 channel scope hookup) as the reference. So perhaps that's where the misunderstanding lies i.e. you are thinking of a split-phase system as being more like a degenerate delta connected system when it is actually more like a degenerate Y connected one.
I do like your newer ScopeMeter though, but for my uses this old dog still hunts.
Fluke does make nice stuff.