Some sluething revealed to me that the first source was really Cartan's La Géométrie des espaces de Riemann, published in 1925 three years before the Lecons sur la geometrie des espaces de Riemann. Looking at the list of Cartan's works, this is the first work on the Riemannian differential geometry. The link for the first source is in the original French; About 60-80% of the text is understandable with only knowledge of English (from my experience, for I do not know French) because the French words for mathematical terms are very similar. For the basis of covariant vectors being $dx$, here are the relevant sections (three sections, close together but with paragraphs in between, images have been posted in French):



This is the source for the statement ""Élie Cartan proposed to use differential coordinates dxi as a convenient basis of 1-forms," though his terminology is outdated in this context (no mention of basis or of differential forms, even though both are being discussed). I believe that in the second source linked (the one given by Mauro ALLEGRANZA) the notation is more modern (being more 'official' than this one) but I haven't checked.
This is where the cute notational game is played. Although the partial derivate operators aren't explicitly shown, because the inner product of the contravariant and covariant bases equals the metric tensor (second image), by definition the contravariant base would be equivalent to the partial derivative operator. I believe that the more modern notation is in the second source linked, but as I have before stated, I haven't checked.
As an addition to this answer, wythagoras translated the French text to English. Note that he had to reform some of the sentences or split them in two sentences.
Citation 1.
An entire system of curvlinear coordinates $u^1$, $u^2$, ... $u^n$ in $n$-dimensional Euclidean space correspond to a determined form of linear elements of the space $$ds^2=\sum_{i,j} g_{ij} du^i du^j$$ where the coefficients $g_{ij}$ are certain functions in $u^1$, $u^2$, ... $u^n$.
The two fundamental problems that arise are the following:
The linear element of the Euclidean space is a priori supposed to give rise to a point such that the curvlinear coordinates can be compared to the Cartesian coordinates.
Find the conditions under which the functions $g_{ij}$ are able to represent the $ds^2$ from the Euclidean space.
Citation 2.
Imagine for each point $M$ the curvlinear coordinates $u^1$, $u^2$, ... $u^n$ a system of cartesian coordinates (R) with the origin on that point. Now consider the point on infinitesimal distance of $M$, namely $M'$, and compare it to $du^1$, $du^2$, ... $du^n$. The coordinate axes are therefore tangents to the line coordinates and the scalar products of the vectors of the coordinates $e_1, e_2, e_n$ is the quantity $g_{ij}$.
Citation 3.
The conditions for integrability of equation (4) provide the other equations. First notice that the first member of equation (4), which we call $Dx^i$, represent the components of motion of a mobile point, of which the coordinates, comparing to (R), would be $x^i$. Likewise the components of the geometric variation of a vector $X^i$ would be: $$DX^i = dX^i + \sum_{k,r} \Gamma^i_{k,r}X^k du^r$$