This seems to have started with Jacobi around 1840, when he re-introduced the popular notation for partial derivatives in De determinantibus functionalibus. He even provides a well-intentioned justification for starting this abuse of notation: he wanted a less ambiguous notation.
While reading Jacobi one should bear in mind, that during his time the $f$ in $f(x)$ was not yet officially called the function. This modern use only started after 1900 and after Frege, Dedekind, Peano and Cantor. (For more on this see Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?.) Prior to 1900 what was called the function was the $f(x)$ or the $y$ in $y=f(x)$, in other words the variable quantity, and not the "rule" $f$. This is also how Jacobi uses the word function below. Beware hence, that what he denotes with $f$ and calls a function, is actually a variable quantity. (In modern terminology: his $f$ is an object of type $\mathbb{R}$ and not of type $\mathbb{R}\to\mathbb{R}$.)
What follows is my translation of the German translation Ueber die Functionaldeterminante, 1941 p.2 ff:
Before I come to the actual subject matter, I will start with some
remarks concerning the notation of partial differentials. And since
this treatise will contain repeated talk about functions, which may or
may not depend from one another, it seems appropriate to also add some
elementary considerations about these.
$$2. $$
To distinguish the partial differentials from the ordinary ones,
hence from those where all variable quantities are seen as functions
of a single variabel, Euler and others put the partial differentials
in between brackets. But since an accumulation of brackets becomes
rather annoying for reading and writing, I have preferred to use the
characteristic $$ d $$ to denote ordinary differentials and the
characteristic $$ \partial $$ for partial differentials. Adopting this
convention rules out misunderstandings. So if $f$ is a function of
$x$ and $y$, I will write
$$ df =\frac{\partial f}{\partial x}dx +\frac{\partial f}{\partial y}dy. $$
Whenever a function contains only a single variable, one may use the
characteristic $d$ or $\partial$ indifferently.
[...]
In order for the partial differentials, of a function which depends on
more than one variable, to be completely determined, it does not
suffice to provide the function to be differentiated and the variable
with respect to which to differentiate; one must moreover express
which quantities remain constant during the differentiation. For
suppose $f$ is a function of $x,x_{1},\dotsc, x_{n}$. Take $n$
arbitrary functions $\omega_{1}, \omega_{2}, \dotsc, \omega_{n}$ of
these variables and consider $f$ as a function of the variables
$x,\omega_{1}, \omega_{2}, \dotsc, \omega_{n}$. Then, when
$x_{1}, \dotsc, x_{n}$ remain constant, the
$\omega_{1}, \omega_{2}, \dotsc, \omega_{n}$ will no longer be
constant with changing $x$, and neither will, when
$\omega_{1}, \omega_{2}, \dotsc, \omega_{n}$ remain constant, the
$x_{1}, \dotsc, x_{n}$ stay constant. The expression $\frac{\partial f}{\partial x}$ will hence assume completely different values,
depending on whether these or those quantities are assumed constant
during differentiation.
(In modern terminology one could rephrase this as follows: the vector field $\frac{\partial}{\partial x}$ associated to a coordinate chart $x,x_1,\ldots,x_n$ on a manifold, does not depend on the coordinate function $x$ alone, but also on the remaining coordinate functions $x_1, \dotsc, x_n$.)
Suppose for example we introduce for a function $f$ of the two
variables $x$ and $y$, an arbitrary function $u$ of $x$ and $y$, as
the second independent variable instead of $y$. Then the differential
that was previously denoted by
$$ \frac{\partial f}{\partial x} $$
will now be expressed as
$$ \frac{\partial f}{\partial x} +
\frac{\partial f}{\partial u}\cdot\frac{ \partial{u}}{{\partial x}},
$$ so that the same signs $\frac{\partial f}{\partial x}$ denote
completely different values, depending on whether $y$ or $u$ are kept
constant, while differentiating $f$ with respect to $x$.
I will therefore in this treatise, whenever partial differentials are
needed, not only express with the statement: $f$ is a function of
$x, x_{1}, \dotsc, x_{n}$, that $f$ depends on these variables, hence
that it remains constant when these are constant, and changes, when
they change — that would equally be valid, if instead of
$x, x_{1}, \dotsc, x_{n}$ any other variables $\omega, \omega_{1}, \dotsc,
\omega_{n}$ functions of these, were introduced as independent
variables — rather when I say, $f$ is a function of the
$x, x_{1} , \dotsc, x_{n}$ I want the following to be understood:
whenever this function is partially differentiated, the
differentiation should occur in such a way, that of these variables
only one changes while the others remain constant.
Further, if the formulas are to be free of ambiguities, the notation
should not only indicate the variable with respect to which the
differentiation is occurring, but also the whole system of
independent variables, the function of which is being differentiated,
so that one may recognise which quantities remain constant during
differentiation. And this is all the more necessary, since it is
unavoidable that in the same calculation or even in one and the same
formula, there appear partial differentials that refer to different
systems of independent variables, for example in the above expression
$$ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial u}\cdot
\frac{\partial{u}}{{\partial x}} $$ in which $f$ is seen as a function
of $x$ and $u$, while $u$ is seen as a function of $x$ and $y$; this
expression was precisely the one $\frac{\partial f}{\partial x}$
transitions into, when $u$ is introduced as independent variable
instead of $y$. But if we write next to the dependent variable all the
independent variables to which the partial differentiation refers,
then this expression can be depicted by the following formula, which
is free of any ambiguity: $$ \frac{\partial f(x,y)}{\partial x} = \frac{\partial f(x,u)}{\partial x} +
\frac{\partial f(x,u)}{\partial u}\cdot
\frac{\partial{u(x,y)}}{{\partial x}}. $$
(To stress: he is using the same notation as function application, but he doesn't want to denote function application with it, he only wants to make explicit which variables are to be kept constant.)
Certainly this notation, as well as every other imaginable notation
that would allow one to completely determine any partial
differentiation from the symbols alone, would become very cumbersome
in more general investigations or more involved formulas, yes even
impracticable, since with higher numbers of independent variables and
more terms it might happen that a formula, which can be expressed in a
single line, takes up a whole page. Certainly one should place the
highest value on a notation that eliminates any ambiguity, and which
makes any formula understandable on its own, without oral
clarifications. But when it was possible without too much disadvantage,
and in view of the enormous and unavoidable verbosity of the notation,
I settled for the shorter notation of differentials, that dispenses
with the specification of the independent variables.
Up to this point one might argue that his notation is justified. But he starts doing it in places where I see no reason for it. For example further down he writes
Let $f, f_1, f_2$ etc. be mutually independent functions of the variables $x, x_1, \ldots, x_n$, and let $x$ be a variable contained in $f$ [...] it follows that the $m+1$ equations
\begin{align}
f(x,x_1,\ldots,x_n)&=\omega\\
f_1(x,x_1,\ldots,x_n)&=\omega_1\\
&\ldots\\
f_m(x,x_1,\ldots,x_n)&=\omega_m
\end{align}
do not determine more than $m+1$ quantities.
Here these $f$ are again variables quantities of type $\mathbb{R}$ and not of type $\mathbb{R}\to \mathbb{R}$, and there is no need to write $(x, x_1, \dotsc, x_n)$ next to the $f$'s in the equations.
I haven't been able to find similar "typing errors" in Bernoulli, Euler, Lagrange, Laplace, Gauss or Cauchy. Even after Jacobi there are several people like Riemann or Peano where I can't find this. Although it is easier to find in the second half of the 19th century. For example in Hermite or Maxwell and later in Felix Kleins lectures on mechanics as well as in Sommerfeld.
But since I have only looked at 1 or 2 works of each of the above authors, this answer is by no means conclusive.
