The original nonsense response is deleted. Below is a letter from Allen Hatcher himself addressing the issue. I wish to apologize for my earlier misunderstanding and stupid remarks.
Let me try to answer the two questions asked by Lorban in the math.stackexchange posting that you linked to. The first question asks about the difference between the formula for T that I give and the formula of Rotman which has an extra term. Both formulas are inductive, assuming one has already defined T for simplices of lower dimension. My formula is
$T\lambda = b_\lambda(\lambda - T\partial\lambda) \tag{$\ast$}$
The right side of this equation can be rewritten as
$b_\lambda(\lambda) - b_\lambda(T\partial\lambda) \tag{$\ast\ast$}$
As explained in my book, the way to understand this geometrically is to think of the simplex $\Delta^n$ as the projection of a prism $\Delta^n \times I$ that has $\lambda$ as its base face and $b_\lambda$ as the barycenter of its top face. Then the first term $b_\lambda(\lambda)$ in the formula $(\ast\ast)$ is the cone on the base of the prism, with the tip of the cone being $b_\lambda$ in the top face, and the second term $b_\lambda(T\partial\lambda)$ is the cone on the sides of the prism, which have already been subdivided by induction.
Rotman's formula is
$T\gamma = b.(\gamma - Sd\gamma - T\partial\gamma) = b.\gamma -b.Sd\gamma - b.T\partial\gamma $
Here $b$ is to be interpreted as the center point of the whole prism, halfway between the top and bottom of the prism, and the three terms in the formula represent the cone on the bottom (this is $b.\gamma$), the cone on the top (this is $b.Sd\gamma$) and the cone on the sides (this is $b.T\partial\gamma$).
I'm not sure what the reason is for doing it Rotman's way, coning to the center of the whole prism rather than to the center of the top. There are other books that do the same thing as Rotman. It seems simpler to do it the way I did because the formulas have fewer terms.
Someone called user32240 attempted an answer to Lorban's question, but the answer contains a fundamental mistake. In the middle of the answer there is an equation
$[a,b]−T(b−a)=(a,b)−(b,b)+(a,a)=(2a−b,a)$
The first equality here is correct, but the second one is wrong. First of all, there should really be square brackets in these formulas to indicate simplices:
$[a,b]−T([b]−[a])=[a,b]−[b,b]+[a,a]$
This is a linear combination of three 1-simplices, two of which are degenerate, $[a,a]$ and [$b,b]$. It makes no sense to try to combine these three 1-simplices into a single 1-simplex $[2a-b,a]$. These are not vectors that we perform vector addition on, they are just combinatorial objects, simplices. The whole point of homology theory is to take formal linear combinations of simplices, not to treat simplices as vectors and add them.
This mistake messes up many of the later formulas as well.
To compute the correct formula for $T([a,b])$ let us denote the barycenter of $[a,b]$ by $c$ (this is $b_\lambda$ in my earlier notation). Then we get
$T([a,b]) = c([a,b] − T([b]−[a])) = [c,a,b]−[c,b,b]+[c,a,a]$
This is a linear combination of three 2-simplices, all of which are actually degenerate since $c$ is the barycenter of $[a,b]$. Geometrically, one can see the three 2-simplices in the prism $\Delta^1 \times I$ (which is just a rectangle) by labeling the two vertices on the left edge by $a$, the two vertices on the right edge by $b$, and the midpoint of the top edge by $c$. There are edges in the interior of the rectangle connecting $c$ to the two ends of the bottom edge, and these two edges subdivide the rectangle into three 2-simplices, $[c,a,b]$, $[c,b,b]$, and $[c,a,a]$.
I hope this long explanation clarifies things.