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\documentclass[12pt, a4paper, openany]{report}

%Disables "ibid" in running text while leaving it activated in footnotes:
% Hinweis: Änderungen an einem Paket sollte man _nach_ dem Laden des Pakets durchführen, sonst bringt das meist eher wenig. Hier bringt es ausnahmsweise etwas, weil etwas neues definiert wird. Dafür wäre es gut, wenn man hier in guter LaTeX-Manier \newcommand statt \def verwenden würde.
\makeatletter
\def\blx@opt@ibidtracker@foot{%
  \let\blx@imc@ifciteibid\blx@ifciteibid@foot
  \let\blx@ibidtracker\blx@ibidtracker@foot
  \let\blx@ibidreset\blx@ibidreset@foot
  \booltrue{citetracker}}

\def\blx@ifciteibid@foot{%
  \ifbool{citetracker}
    {\iftoggle{blx@footnote}
       {\blx@imc@iffieldequals{entrykey}\blx@lastkey@text}
       {\@secondoftwo}}
    {\@secondoftwo}}

\def\blx@ibidtracker@foot{%
  \ifbool{citetracker}
    {\iftoggle{blx@footnote}
       {\global\let\blx@lastkey@text\abx@field@entrykey}
       {\global\let\blx@lastkey@foot\abx@field@entrykey}}
    {}}

\def\blx@ibidreset@foot{%
  \iftoggle{blx@footnote}
    {\global\undef\blx@lastkey@text}}
    {}
\makeatother

\usepackage[backend=biber, style=authortitle-dw, useprefix=true, acronyms=true, ibidtracker=foot]{biblatex}
\usepackage[left=2.5cm, top=2.5cm, right=2.5cm, bottom=2cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[ngerman]{betababel}
\usepackage{csquotes}

\makeatletter
\renewcommand*{\mkbibacro}[1]{% Definition aus Abschnitt 7.2 der Anleitung zu authortitle-dw kopiert und \newcommand durch \renewcommand ersetzt.
  \ifcsundef{\f@encoding/\f@family/\f@series/sc}
    {#1}
    {\textsc{#1}}}% \MakeLowerCase entfernt.
\makeatother

\begin{filecontents}{\jobname.bib}
@book{Hegel,
  author      = {Hegel, Georg Wilhelm Friedrich},
  title       = {Enzyklopädie der philosophischen Wissenschaften III},
  options     = {acronym=true},
  shorthand   = {Enz III},
  }
\end{filecontents}
\addbibresource{\jobname.bib}

\begin{document}

Text Text Text \parencite[10]{Hegel}. Text Text Text \parencite[10]{Hegel}.

\end{document}

\documentclass[12pt, a4paper, openany]{report}

%Disables "ibid" in running text while leaving it activated in footnotes:

\makeatletter
\def\blx@opt@ibidtracker@foot{%
  \let\blx@imc@ifciteibid\blx@ifciteibid@foot
  \let\blx@ibidtracker\blx@ibidtracker@foot
  \let\blx@ibidreset\blx@ibidreset@foot
  \booltrue{citetracker}}

\def\blx@ifciteibid@foot{%
  \ifbool{citetracker}
    {\iftoggle{blx@footnote}
       {\blx@imc@iffieldequals{entrykey}\blx@lastkey@text}
       {\@secondoftwo}}
    {\@secondoftwo}}

\def\blx@ibidtracker@foot{%
  \ifbool{citetracker}
    {\iftoggle{blx@footnote}
       {\global\let\blx@lastkey@text\abx@field@entrykey}
       {\global\let\blx@lastkey@foot\abx@field@entrykey}}
    {}}

\def\blx@ibidreset@foot{%
  \iftoggle{blx@footnote}
    {\global\undef\blx@lastkey@text}}
    {}
\makeatother

\usepackage[backend=biber, style=authortitle-dw, useprefix=true, acronyms=true, ibidtracker=foot]{biblatex}
\usepackage[left=2.5cm, top=2.5cm, right=2.5cm, bottom=2cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[ngerman]{betababel}
\usepackage{csquotes}

\begin{filecontents}{\jobname.bib}
@book{Hegel,
  author      = {Hegel, Georg Wilhelm Friedrich},
  title       = {Enzyklopädie der philosophischen Wissenschaften III},
  options     = {acronym=true},
  shorthand   = {Enz III},
  }
\end{filecontents}
\addbibresource{\jobname.bib}

\begin{document}

Text Text Text \parencite[10]{Hegel}. Text Text Text \parencite[10]{Hegel}.

\end{document}