Roshan Kakiya,
While I share your fascination with the various forms of equal temperament, I should point out that there is established literature on this subject that you might want to investigate before considering yourself ready to participate in its discussion.
You write:
"The inharmonicity of a piano's strings needs to be dealt with in order to make a piano sound in tune."
True.
"Piano tuners deal with the inharmonicity of a piano's strings by stretching Octaves."
Yes, and there are established systems in place to quantify that stretch.
"In theory, the standard version of 12-Tone Equal Temperament corrects the Pythagorean Comma by narrowing all 12 Pure Fifths whilst preserving the purity of all 7 Octaves."
True, but the name is '12-tone to the pure octave equal temperament.'
"In practice, the Octaves are stretched which means that the Octaves will not be pure in reality."
In practice, regardless of whether there is stretch or no stretch, there are no pure octaves on the piano because of the change in inharmonicity through the scale, making it impossible for partials to align when tuning octaves. Beyond that, one should keep in mind that correcting octaves (or any interval) for inharmonicity is not necessarily considered to be 'stretch'. By correcting for inharmonicity and not stretching beyond that of correcting for inharmonicity, it is thoroughly possible to tune pure octave equal temperament on real pianos; this has been demonstrated repeatedly.
"This means that the theory of the standard version of 12-Tone Equal Temperament does not match the practice of stretching Octaves."
The mathematical model of equal temperament assumes zero inharmonicity. Real pianos have inharmonicity and the method for applying the model of equal temperament to the inharmonic piano is one of humanity's great endeavors.
"In theory, the alternative version of 12-Tone Equal Temperament corrects the Pythagorean Comma by widening all 7 Octaves whilst preserving the purity of all 12 Pure Fifths.
Yes, but the name of this equal temperament is '7-tone to the pure fifth equal temperament.'
Pure octave equal temperament is 'narrow' equal temperament, while pure 5th equal temperament is relatively 'wide.' In addition there are an infinite number of equal temperaments with widths between that of pure octave ET and pure 5th ET. The desired amount of stretch, then, may be 'quantized' by executing a chosen width of equal temperament. Pianos may be tuned to several of these widths of equal temperament with great precision with the aid of the Verituner electronic tuning device.
"This means that the theory of the alternative version of 12-Tone Equal Temperament does match the practice of stretching Octaves."
Not really. The desired amount of stretch must match the width of equal temperament. In the 21st century stretch and width of equal temperament are the same thing, and can function independently of inharmonicity.
"The inharmonicity of a piano's strings increases as we move up the keyboard:"
True.
"Therefore, not all octaves will be stretched by the same amount."
No. Don't confuse the effect of inharmonicity with the choice to tune an equal temperament that is wider than pure octave. The correction needed for inharmonicity will increase as one moves higher in the scale, but the overlaid width of equal temperament can still be consistent.
"The alternative version of 12-TET only contains octaves of equal size in order to theoretically fulfil the definition of equal temperament (every semitone must be equal in terms of size)."
And don't confuse the mathematical model of equal temperament with the inharmonicity-perturbed version of equal temperament that we lay upon pianos.
"In practice, the alternative version of 12-TET can be used as a framework for controlling inharmonicity. This is because it naturally accounts for the stretching of octaves. The size of the octaves can be adjusted in accordance with the specific inharmonicity of a piano."
Yes, a chosen width of equal temperament (that is, stretch) may very well help the tuner deal with the effects of inharmonicity, but the subject is more complex that you make it here.
May I suggest you read Mark Cerisano's recently published book on piano tuning theory and perhaps my own series of articles on 21st century tuning style in the Piano Technicians Journals.
Kent