I have recently encountered this question. The working I saw first translated the graph one unit in the positive $x$-direction, removed the part of the graph in negative $x$-region, and then do the reflection of the graph in positive $x$-region into the negative $x$-region. This working is what I have in mind too.
However, the answer first changed the graph of $y=f(x)$ to graph of $y=f(|x|)$. then do the translation. If this is the case, will it be right to say that the whole process of removed the part of the graph in negative $x$-region, and then do the reflection of the graph in positive $x$-region into the negative $x$-region is replacing the $x$ in $y=f(x)$ to $|x|$? I will admit this is the first time I see such combination of transformations, so it is always just $y=f(x)$ to $y=f(|x|)$ (maybe some changes done to $y$).
You are correct, given any function $f(x)$, you get the graph of $f(|x|)$ by taking the graph of $f$, deleting the graph where $x<0$, then reflecting the graph where $x>0$ into the region you deleted.
To understand the answer described in your second paragraph, think of $f(|x-1|)$ as $g(x-1)$ where we define $g$ as the new function $g(x):=f(|x|)$. So you can plot $f(|x-1|)$ in two steps:
It's important to distinguish the function $f(|x-1|)$ from the function $f(|x|-1)$. Notice these functions are not the same, since the order of operations is different. The approach you described in your first paragraph appears to be plotting $f(|x|-1)$ instead of $f(|x-1|)$.