In J. C. Baez's February 9, 2000 entry on his website it is claimed that the process of coordinatizing a Desarguian projective plane (i.e. obtaining a skew field/division ring in the style of the Greeks) is inverse, up to isomorphism, to the usual $K \mapsto K\mathbf{P}^2$ construction.
I cannot find a proof of this obvious but probably quite technical fact after an admittedly inexhaustive survey of the provided references.
Does someone have a reference for this? Many thanks.
On
Disclaimer: This is not really an answer but it might help.
One thing I know is that for affine planes the process of coordinatizing is fully described in the book
From Affine to Euclidean Geometry: An Axiomatic Approach by Wanda Szmielew
It is done for different classes of affine planes (Desarguesian, Pappian, general) and starts from synthetic axiomatization (in the style of the Greeks).
When it comes to projective planes I would try to invent the proof with the following idea:
Given a projective plane, removing one line produces an affine plane. If the projective plane is Desarguesian (Pappian) so is the produced affine plane. Desargusian affine plane is isomorphic to $F^2$, where $F$ is a division ring (check Szmielew book for the proof). From here I don't know exactly how to proceed (that's why it is a partial answer).
Beutelspacher & Rosenbaum, Projective Geometry: From Foundations to Applications, pg 117 proves the following theorem, which is what I think you want.
As suggested by @Kulisty the proof builds on a representation of affine spaces.
See also Mandelkern, Constructive Coordinatization of Desarguesian Planes for the remarks and references given in the introduction.
The latter leads to Artin, Geometric Algebra, Chapter 2 and Hilbert's Foundations of Geometry, Chapter V.
According to Mandelkern, "the classical theory of Desarguesian planes and their coordinatization [...] originated with D. Hilbert" and Artin gives a modern presentation.