Convention in electrochemical cell notation
Good question. You can remember the cell notation with the mnemonic &quot;reduction on the right.&quot;
So, on the right hand side, you should reduction process and on the left hand side you should show the oxidation process.
For example, for a Daniel cell, one can write
$$\ce{Zn|Zn^{2+}, SO_4^{2−}}||\ce{SO_4^{2−},Cu^{2+}}|\ce{Cu}$$
Now coming to your cell, $$\ce{Pt,Cl_2}|\ce{Cl^-(aq)}||\ce{Ag^+(aq)}|\ce{Ag}$$ is indeed written incorrectly because the writer seems to have the misconception that even in the cell notation, half cells must be written as a reduction.