After having read Katz' article, I must say I am not convinced and find that the standard interpretation, namely that of Cauchy making a mistake in 1821 and failing to acknowledging it or correcting it properly in 1853 is closer to the truth. In other words, even after reading your paper, I see nothing more than meets the eye.

One of your main point is the word "toujours" (always) which appears in the 1853 version of Cauchy's theorem, but not in the 1821 version. Quoting your paper, the 1821 version says

When the various terms of series $u_0 +u_1 +u_2 + \dots +u_n + u_{n+1} + \dots$ are functions of the same variable $x$, continuous with respect to this variable in the neighborhood of a particular value for which the series converges, the sum s of the series is also a continuous function of $x$ in the neighborhood of this particular value.

(I would have liked to see the French version, by the way).

The 1853 version is:

Théorème 1. Si les différents termes de la série $$u_0,u_1,u_2,\dots,u_n,u_{n+1},\dots \ \ (1)$$ sont des fonctions de la variable réelle $x$, continues, par rapport à cette variable, entre des limites données; si, d’ailleurs, la somme $u_n +u_{n+1} + \dots + u_{n′−1}$ devient toujours infiniment petite pour des valeurs infiniment grandes des nombres entiers $n$ et $n′ > n$, la série (1) sera convergente et la somme $s$ de la série sera, entre les limites données, fonction continue de la variable $x$.

You interpret "toujours" as meaning "for real (archimedean) and for infinitesimal values of the variable $x$". But I note that it is more natural to interpret it simply as meaning "for all real (archimedean) values of $x$". This interpretation would be enough to make the 1853 statement different, precisely with a stronger hypothesis, than the 1821 statement, for plainly the 1821 statement requires only the convergence of the series for a particular value $x_0$ (and the continuity of the $u_n$ on a neighborhood of $x_0$) to conclude the continuity of the sum $s$ at $x_0$. Thus we would have two statements of Cauchy's theorem, which both happen to be false.

The second important point of your argument is the discussion of Cauchy's treatment of a potential counter-example related lo Abel's objection in section 2.3. Cauchy claims that this is not a counter-example to his 1853 theorem because it fails some hypothesis. But here, since you give no quotation of Cauchy, it is impossible to know if Cauchy's arguments support your interpretation or are simply mistaken.