I think the issue here is your understanding of physics and stationary waves.
What you have when a harmonic is plays is indeed a "stationary wave".
Where the finger is placed causes "node" to form which in this type of wave is a location of no movement. This would be where the 1/3 mark is. An antinode is the location of the greatest movement where the crests are.
The wavelength of that wave is equal to the side you labeled "Long".
The short side does not have a wavelength of its own. It is part of the whole wave. The short side is not any interval on its own.
This site explains it in greater detail.
Let's look at it using actual numbers
Take A4 = 400Hz
and v = fλ
- v = speed of sound
- f = frequency of the sound
- λ = wavelength
We are idealizing so we are ignoring the effect of tension on frequency of a string by considering the speed of sound in the air.
using a little algebra we see
f = v / λ
since the speed of sound in air is constant we then see that
f is inversely proportional to λ, meaning when the frequency is multiplied by a number the wavelength is divided by the same.
If we manage to produce an A4 on a string 1 unit long. In order to double the frequency (or produce a P8 over the open string) we have to divide the length of the string by two.
to get two octaves above we can divide by 4.
Now what happens when we divide the string into thirds?
Well since it is inversely proportional we get a frequency of 3 times higher.
E6 = 1320Hz. This is the pitch when played as a harmonic. In which case the new wavelength is actually 2/3 of the open string.
Remember the wavelength for a stationary wave is not the distance between two nodes but double that.
So your errors are as follows:
You are considering the short section as a separate "wave" with it's own length. it isn't.
The wavelength of the wave is actually the the length of the section that you labled Long.
There is no harmonic that is not at least an octave away from the fundamental.
I understand why one might thing a dividing by 3 would produce a P5. When a string is stopped only one side is allowed to vibrate. In addition, no node is produced. So if someone stops a string and plays it one third the distance from the nut (the point labled 1/3 in the image) then resulting wavelength is actually double that of the harmonic on the third. And remember if we double the length we half the frequency so the resulting pitch is E5.
However if were stop the string at the node nearest to the right of the string in your diagram, we get the pitch that matches the harmonic.