A proof can be extracted from Cartan's "On manifolds with projective connections." (Translated by D. H. Delphenich). In section VII Cartan defines a projective connection in the projectivised tangent bundle ("manifold of elements") with connection one-form: $$ \omega = \begin{pmatrix} \omega_0^0&\omega^1&\omega^2\\ \omega_1^0&\omega_1^1&\omega_1^2\\ \omega_2^0&\omega_2^1&\omega_2^2\\ \end{pmatrix} $$ After normalising the connection he arrives at the curvature two-form $$ \Omega = \begin{pmatrix} \gamma&0&0\\ \Omega_1^0&\gamma&0\\ \Omega_2^0&\Omega_2^1&\gamma\\ \end{pmatrix}, $$ where $$ \begin{aligned} \Omega_2^1 &= a\omega^2\wedge\omega_1^2,\qquad a=-\frac{1}{6}\frac{\partial^4f}{\partial y'^4}\\ \Omega_1^0&=b\omega^1\wedge\omega^2,\qquad b=\text{complicated function}\\ \Omega_2^0&=\text{complicated two-form}\\ \gamma&=\Omega_0^0=\Omega_1^1=\Omega_2^2=\text{complicated two-form}.\\ \end{aligned} $$ In section 23 Cartan then states the dual connection with connection one-form $$ \varpi = \begin{pmatrix} \varpi_0^0&\varpi^1&\varpi^2\\ \varpi_1^0&\varpi_1^1&\varpi_1^2\\ \varpi_2^0&\varpi_2^1&\varpi_2^2\\ \end{pmatrix}=\begin{pmatrix} \omega_2^2&\omega_2^1&\omega^2\\ \omega_2^1&\omega_1^1&\omega^1\\ \omega_2^0&\omega_1^0&\omega_0^0\\ \end{pmatrix} $$ which is simply the primal connection transposed with respect to the antidiagonal and the same goes for the the curvature two-form: $$ \Pi = \begin{pmatrix} \gamma&0&0\\ \Pi_1^0&\gamma&0\\ \Pi_2^0&\Pi_2^1&\gamma\\ \end{pmatrix} = \begin{pmatrix} \gamma&0&0\\ \Omega_2^1&\gamma&0\\ \Omega_2^0&\Omega_0^1&\gamma\\ \end{pmatrix}, $$ so that we have $$ \begin{aligned} \Pi_1^0=\Omega_2^1\\ \Pi_2^1=\Omega_1^0 \end{aligned} $$ We have that $\Omega_2^1=0$ is the condition that the right-hand side of the differential equation is a polynomial in $y'$ with degree at most 3 and the condition $\Omega_1^0=0$ is then "infinitesimal desargueness". In the dual connection these two conditions are swapped.
It might be argued that "infinitesimal nondesargueness" should really mean that any of $\Omega_2^1$ or $\Omega_1^0$ is non-vanishing since they are both obstructions to the construction of homogenous coordinates which then gives a projective structure on the $x,y$-plane with the solution curves as lines. The availability of homogenous coordinates is equivalent to the Desarguesian property of a projective plane.
Here's a couple of blog posts discussing these things: