How to prevent latex from removing the margin from a PDF image

I have a chart in PDF format, which had been cropped to have no margin using pdfcrop

Months later, I realized that I need to add X- and Y- axis labels. Because I don't have the original script to generate the figure, I edited the PDF by adding a narrow margin to the left- and bottom corners:

pdfcrop --margin '20 0 0 20' fig_tight_cropped.pdf fig_with_axis_labels.pdf

And then edited the PDF and added the X and Y labels (using Mac's preview tool)

However, when I put them in a latex file, PDFLatex makes them appear exactly the same and ignores the added margin in the second file.

Here is the minimum working example:

\documentclass{article}
\usepackage{graphicx}

\begin{document}

\begin{figure}[h!]
\includegraphics{fig_tight_crop.pdf}
\includegraphics{fig_with_axis_labels.pdf}
\end{figure}

\end{document}

And the output looks like this:

How can I make the margins appear? Why does it disappear in the first place?

P.S. Here are the figures with and without the margin:

fig_tight_cropped.pdf

fig_with_axis_labels.pdf

You can add annotations to an image using tikz.

\documentclass{article}
\usepackage{graphicx}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
\pgftext{%
\includegraphics[width=\textwidth]{fig_tight_cropped}%
}%
\node[xshift=-6.5cm,rotate=90] {$C_d$};
\node[yshift=-4.4cm] {$C_x$};
\end{tikzpicture}

\end{document}

The tikz-imagelabels package could help in placing annotations:

\documentclass{article}
\usepackage{tikz-imagelabels}

\imagelabelset{
image label distance=-7mm,
image label back=none,
image label text=black}

\begin{document}

\begin{annotationimage}{width=\textwidth}{fig_tight_cropped}
\draw[image label = {$C_x$ at south}];
\draw[image label = {\rotatebox{90}{$C_d$} at west}];
\end{annotationimage}

\end{document}

The first solution gives you a lot more possibilities. For example you can add a coordinate system and other annotations wherever you like:

\documentclass{article}
\usepackage{graphicx}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
\node [anchor=south west] at (0,0) {\includegraphics[width=\textwidth]{fig_tight_cropped}};
\draw [very thick,draw=red,->] (0,0) -- (0,7) node[below left,midway] {$C_x$};
\draw [very thick,draw=red,->] (0,0) -- (7,0) node[midway, below left] {$C_d$};
\node at (5,6) {An annotation here};
\end{tikzpicture}

\end{document}