Where did this famous Planetary Precession Formula come from?
I do not know where that formula was first published in full, but Oppenheim at least does something very close to it. First, keep in mind some of the relevant symbols in Oppenheim, though they're rather standard:
$$\def\anode{☊}\begin{eqnarray*}
k &amp;=&amp; \small\sqrt{G} = \small\text{Gaussian gravitational constant}\\
\anode &amp;=&amp; \small\text{longitude of the ascending node}\\
\omega &amp;=&amp; \small\text{argument of perihelion}\\
\varpi &amp;=&amp; \omega+\anode = \small\text{longitude of perihelion}\\
n &amp;=&amp; \small{k\sqrt{(m_0+m_1)/a^3}} = \small\text{mean motion}
\end{eqnarray*}$$
We can see that if we massage the notation, the proportionality we're looking for is equivalently stated as:
$$\frac{\pi m}{Lc^2} \leadsto \frac{\pi GM}{a(1-e^2)c^2} \leadsto \frac{\pi n^2a^2}{(1-e^2)c^2}\text{.}$$
Since the mean motion is $n\equiv 2\pi/T$, where $T$ is the orbital period,
$$\frac{n^3a^2}{c^2} = 2\pi\frac{n^2a^2}{c^2}\frac{1}{T}\text{,}$$
which means that if we want to talk about a perihelion advance of $\delta\varpi \propto \frac{\pi n^2a^2}{(1-e^2)c^2}$ per orbit, it is equivalent to talk about a term in the form
$$\frac{\mathrm{d}\varpi}{\mathrm{d}t} \propto \frac{n^3a^2}{(1-e^2)c^2} = \frac{n^3a^2}{c^2}\left(1+e^2+\mathcal{O}(e^4)\right)\text{.}$$
I cannot find where, if indeed anywhere, Oppenheim considers the missing factor of $(1-e^2)$ as belonging to the anomalous precession, but otherwise the formula is definitely there. I suspect that he only uses a circular orbit approximation, $e^2\approx 0$, because it's nowhere to be found in Sec. IV (Weber's 1846 theory) and performing his calculation (sans $1-e^2$) gives me $\delta\varpi = 13.72''$ per century for Mercury, in good agreement with his stated result of $\delta\varpi = 13.65''$, whereas putting in a factor of $(1-e^2)$ by hand gives $\delta\varpi = 14.32''$ instead.

Perhaps Oppenheim did not consider it explicitly and the author of the MathPages took it as obvious that the eccentricity factor should be there. Or perhaps there's a side comment in the text that I don't see; unfortunately, I'm nowhere fluent enough in German to understand much of what's going on.