In dihedral group $D_5$, there are five reflections: $s, sr, sr^2, sr^3, sr^4.$
Q 1. The reflections are shown at: https://math.stackexchange.com/a/490089 to occur around the vertex labelled $1$ only, though had read some alternative way elsewhere.
It further correctly states: $s_0 : (25)(34)$ $$s_0=\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 1 & 5 & 4 & 3&2\end{pmatrix}$$
Q.2. A bigger issue is that it shows the product of transpositions to be:
$s_1= TR : (12)(53)= $ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 2 & 1 & 5 & 4&3\end{pmatrix}$$
But, when try to compose $s_1= TR$ find:
$R =(51234)=(12345)=$ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 2 & 3 & 4 & 5&1\end{pmatrix}$$ $T =(25)(34)=$ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 1 & 5 & 4 & 3&2\end{pmatrix}$$ $TR =(15)(24)=$ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 5 & 4 & 3 & 2 &1\end{pmatrix}$$
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Similarly, have issues with all other reflections' transpositions :
$s_2= TR^2 : (13)(45)$ $$s_2=\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 3 & 2 & 1& 5&4\end{pmatrix}$$
$R^2 =(13524)=$ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 3 & 4 & 5 & 1&2\end{pmatrix}$$ $T =(25)(34)=$ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 1 & 5 & 4 & 3&2\end{pmatrix}$$ $TR^2 =(14)(23)=$ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 4 & 3 & 2 & 1 &5\end{pmatrix}$$
$s_3= TR^3 : (14)(23)$ $$s_3 =\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 4 & 3 & 2 & 1&5\end{pmatrix}$$
$R^3 =(14253)$ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 4 &5 & 1&2&3 \end{pmatrix} $$ $T =(25)(34)=$ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 1 & 5 & 4 & 3&2\end{pmatrix}$$ $TR^3 =(13)(45)=$ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 3 & 2& 1 & 5 &4 \end{pmatrix}$$
$s_4 = TR^4: (15)(24)$ $$s_4 =\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 5 & 4 & 3 & 2&1\end{pmatrix}$$
$R^4 =(15432)=$ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 5 & 1 & 2 & 3&4 \end{pmatrix}$$ $T =(25)(34)=$ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 1 & 5 & 4 & 3&2 \end{pmatrix}$$ $TR^4 =(12)(35)=$ $$\begin{pmatrix} 1 & 2 & 3 &4 &5\\ 2 & 1 & 5 & 4 &3 \end{pmatrix}$$
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Edit:
My $s_1$ equals link's $s_4,$ and vice-versa.
My $s_2$ equals link's $s_3,$ and vice-versa.
The reflections in the link are all there, as you can see. The list $s,sr,sr^2,sr^3,sr^4$ just happens to give them in a different order.
The key is that if you rotate one more time, and then do that same reflection, $s$, the result is a reflection in a different axis. And you can see that in the picture.
It's actually quite a trivial situation. Because the axis gets rotated.
(Remember the pentagon always lands on itself. They're called symmetries. So, if you will, the original pentagon stays fixed underneath. )