Doing matrix multiplication with $\lceil n^3 / \log n \rceil$ processors in $2\log n$ steps by Brent's principle
Yes, you are correct while all textbooks and lecture notes you and I have found so far are wrong in claiming &quot;doing matrix multiplication with $\lceil n^3 / \log n \rceil$  processors in  at most $2\log n$ steps by Brent's principle&quot;
There is hardly anything valuable to add to your meticulous calculation. Anyway, here is an example that specializes your calculation.
An example
Let $n=16$. We have two $16\times16$ matrices, $(A_{ij})$ and $(B_{ij})$ with $\left\lceil \dfrac{n^3}{\log_2n}\right\rceil=\left\lceil\dfrac{16^3}{\log_2 16}\right\rceil=1024$ processors.
It takes 4 parallel steps to compute all the $n^3=4096$ products of the entries $P_{ijk}=A_{ik} \cdot B_{kj}$ for $i,j,k \in \{1,2,\cdots,16\}$ with $1024$ processors.
How to compute $n^2=256$ sums of the $n=16$ numbers $P_{i1j}+ \ldots + P_{i16j}$ for $i,j\in\{1,2,\cdots,16\}$?
We first compute $256\times 8=2024$ sums of two neighbors $S_{ikj}=P_{i(2k-1)j}+P_{i(2k)j}$ for $i,j\in\{1,2,\cdots,16\}$, $k\in\{1,2,\cdots, 8\}$. It takes 2 parallel steps for 1024 processors, where $2$ is $\dfrac{\dfrac{n^3}2}{\dfrac{n^3}{\log_2n}}=\dfrac{\log_2n}2&gt;1$.
There are $256 \times 4 = 1024$, $256 \times 2 = 512$, $256 \times 1 = 256$ sums of two numbers to calculate, respectively. Each of them will take one parallel step because of the serial dependency. We need $3 =\log_2n-1$ parallel steps.
So we need $4 + 2 + 3 = 9$ parallel steps in total, which is bigger than $\lceil 2\log_2 n\rceil = 8$.
Corrected statement
Multiplication of two $n\times n$ matrices can be done in no more than $3\log_2 n+1$ parallel steps with $\lceil n^3 / \log_2 n \rceil$ processors by Brent's principle.
Brent's principle
Here is one version of Brent's principle that applies to current situation.
(Brent’s principle): An algorithm involving $t$ time steps and performing a total of $m$ operations can be executed by $p$ processors in no more than $t + \lceil(m – t) / p\rceil$ time steps in PRAM.
In particular, we have $m=n^3+ n^2(n-1)$ operations and $t=1+\lceil \log_2n\rceil$ time steps for matrix multiplication. If we have $p=\lceil n^3 / \log_2 n \rceil$ processors, the upper limit of parallel steps given by Brent's principle,  $t + \lceil(m – t) / p\rceil$ is $3\lceil\log_2 n\rceil-1$ or $3\lceil\log_2 n\rceil$  or $3\lceil\log_2 n\rceil+1$.