The source for this answer to the question you link includes, at the top,

 $\newcommand{\Ket}[1]{\left|#1\right>}$

while this other answer includes, between "the three vectors" and the first displayed equation,

 $\newcommand{\ket}[1]{|#1\rangle}$

So on that page both $\ket\psi$ and $\Ket\psi$ do what you want, but the spacing for \Ket is better because that author used \left and \right.

Moral of story: the definition is brief, so include it yourself if it makes your life easier.

The source for this answer to the question you link includes, at the top,

 $\newcommand{\Ket}[1]{\left|#1\right>}$

while this other answer includes, between "the three vectors" and the first displayed equation,

 $\newcommand{\ket}[1]{|#1\rangle}$

So on that page both $\ket\psi$ and $\Ket\psi$ do what you want, but the spacing for \Ket is better because that author used \left and \right.

Moral of story: the definition is brief, so include it yourself if it makes your life easier.