Wieso postest Du den Code nicht direkt? Extra eine Datei herunterladen zu müssen, empfinde ich persönlich immer als etwas umständlich...

Ich würde das anders machen:

Code:
\documentclass{article}

\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{graphicx}

\usepackage{mdframed}
\makeatletter
\newmdenv[
  frametitleaboveskip=6mm,
  frametitlebelowskip=\smallskipamount,
  innertopmargin=3mm,
  innerbottommargin=5mm,
  leftmargin=3mm,
  rightmargin=3mm,
  frametitle=\textbf{Definition\definition@name},
]{definition@inner}
\newenvironment{definition}[1][]
  {%
    \if\relax\detokenize{#1}\relax
      \def\definition@name{}%
    \else
      \def\definition@name{:\, #1}%
    \fi
    \definition@inner
    \renewcommand*\thempfootnote{\@arabic\c@mpfootnote}%
  }
  {\enddefinition@inner}

\usepackage{footnote}
\makesavenoteenv{definition@inner}
\makeatother
\begin{document}

\begin{definition}[Minimum Example]
  This is a footnote within an fbox-environment.\footnote{Here, I'd like to
    have a definition.} And now I have this crazy idea of a second footnote
  within the fbox-environment.\footnote{It works with addtocounter and
    stepcounter.}
\end{definition}

\end{document}
Insgesamt fände ich es aber besser, wenn die Fußnoten im Kasten blieben. Das wäre auch leichter zu erhalten:

Code:
\documentclass{article}

\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{graphicx}

\usepackage{mdframed}
\makeatletter
\newmdenv[
  frametitleaboveskip=6mm,
  frametitlebelowskip=\smallskipamount,
  innertopmargin=3mm,
  innerbottommargin=5mm,
  leftmargin=3mm,
  rightmargin=3mm,
  frametitle=\textbf{Definition\definition@name},
]{definition@inner}
\newenvironment{definition}[1][]
  {%
    \if\relax\detokenize{#1}\relax
      \def\definition@name{}%
    \else
      \def\definition@name{:\, #1}%
    \fi
    \definition@inner
  }
  {\enddefinition@inner}

\begin{document}

\begin{definition}[Minimum Example]
  This is a footnote within an fbox-environment.\footnote{Here, I'd like to
    have a definition.} And now I have this crazy idea of a second footnote
  within the fbox-environment.\footnote{It works with addtocounter and
    stepcounter.}
\end{definition}

\end{document}


Grüße