There seem to be three (four) macros at play here (judging from your example code).
First, \cleardoublepage, which is the standard LaTeX command for clearing pages in two-sided documents. It is used before the \part in your document.
Second (and third), \clearforchapter, which is memoir's command for using the correct page for a chapter heading. However this macro is defined to be either \cleartorecto, \clearpage, or \cleartoverso if any of openright, openany, or openleft was used, respectively (the default is openright). A less wrong change would be to add the intentionally-empty page text in these macros, so if you happen to change the open... option, your document won't be messed up. The answer you linked to happen to work because OP didn't use any option, so openright was used by default (and the redifinition was done based on \cleartorecto).
Fourth, \partpageend, which is used by memoir after a \part.
I also added, for convenience, a \clearpagetext command which shows from which of these macros it was called, just for bookkeeping. You can (and should :-) remove that from your document. That said, here is the code:
\documentclass[openright]{memoir}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\makeatletter
\def\clearpagetext#1{%
\vfill
\begin{center}%
This page was intentionally left blank by \texttt{\string#1}.
\end{center}%
\vfill}
\if@openleft
\def\cleartoverso{\clearpage\if@twoside
\ifodd\c@page\hbox{}%
\clearpagetext\cleartoverso% <--
\thispagestyle{cleared}%
\newpage\if@twocolumn\hbox{}\newpage\fi\fi\fi}
\else
\if@openright
\def\cleartorecto{\clearpage\if@twoside
\ifodd\c@page\else\hbox{}%
\clearpagetext\cleartorecto% <--
\thispagestyle{cleared}%
\newpage\if@twocolumn\hbox{}\newpage\fi\fi\fi}
\fi
\fi
\def\cleardoublepage{\clearpage\if@twoside
\ifodd\c@page\else\hbox{}%
\clearpagetext\cleardoublepage% <--
\thispagestyle{cleared}%
\newpage\if@twocolumn\hbox{}\newpage\fi\fi\fi}% (Yes, this is the same as \cleartoverso)
\def\partpageend{\afterpartskip
\ifm@mnopartnewpage
\else
\if@twoside
\if@openright
\null
\clearpagetext\partpageend% <--
\thispagestyle{afterpart}%
\newpage
\fi
\fi
\fi
\if@tempswa
\twocolumn
\fi}
\makeatother
\begin{document}
Text before a part
\part{bob}
\chapter{the}
\chapter{cool}
\chapter{duck}
\end{document}
The document structure will look like this (behold my fabulous ASCII art):
|---------------| |---------------|
| Text before | | This page was |
| a part | | intentionally |
| | → | left blank by |
| 1 | | \cleardoublepage. |% <-- Overfull \hbox
|---------------| ↙ |---------------|
|---------------| |---------------|
| Part I | | This page was |
| Bob | | intentionally |
| | | left blank by |
| 3 | | \partpageend. |
|---------------| |---------------|
|---------------| |---------------|
| Chapter 1 | | This page was |
| the | | intentionally |
| | | left blank by |
| 5 | | \cleartorecto |
|---------------| |---------------|
|---------------| |---------------|
| Chapter 2 | | This page was |
| cool | | intentionally |
| | | left blank by |
| 7 | | \cleartorecto |
|---------------| |---------------|
|---------------|
| Chapter 3 |
| duck |
| |
| 9 |
|---------------|
\cleartorecto,\cleartoverso, they both use the cleared page style. BTW: why do people want to addthis page left intentionally blank? I've never understood that. – daleif May 29 '19 at 08:59