footcite equals texcite with special format (parenthesis around year and given name initials)

Question

I need to use a special footcite and textcite format for my master thesis. They should both look like
Cite with footnote and textcite.footnote{Vgl. textcite{A01}, S. 23.}
Cite with footcite.footcite[Vgl.][S. 23]{A01}

I tried different solutions for the initials in footcite, but they would corrupt the output of textcite. Here is the version without the initials.

Minimal working example:
documentclass{article}
usepackage[utf8]{inputenc}

usepackage[
backend=biber,
bibencoding=utf8,
style=ext-authoryear,
giveninits=true,
dashed=false,
doi=false,
date=year,
isbn=false,
uniquename=init]{biblatex}
usepackage{filecontents}

begin{filecontents}{jobname.bib}
@article{A01,        
  author = {Author, A.},
  year = {2001},
  title = {Alpha},
  journaltitle = {A Journal},
  volume = {1},
  number = {1},
}
end{filecontents}
addbibresource{jobname.bib}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% footnotestyle format
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Parathenthesis around year in footcite
DeclareInnerCiteDelims{footcite}{bibopenparen}{bibcloseparen}

% indentation
usepackage[hang,bottom,stable]{footmisc}
setlength{footnotemargin}{5mm}
interfootnotelinepenalty=10000

% given name initials for footcite and textcite
% ???

begin{document}
Cite with footnote and textcite.footnote{Vgl. textcite{A01}, S. 23.}
Cite with footcite.footcite[Vgl.][S. 23]{A01}
printbibliography
end{document}

moewe · Accepted Answer

You can modify the labelname name format. The original definition can be found in biblatex.def (ll. 955-981 in v3.16). (Cf. also How to include first initials when using authoryear referencing format (harvard) with biblatex?)
documentclass[ngerman]{article}
usepackage[T1]{fontenc}
usepackage{babel}
usepackage{csquotes}

usepackage[
  backend=biber,
  style=ext-authoryear,
  giveninits=true,
  uniquename=init,
  dashed=false,
  date=year,
  doi=false,
  isbn=false,
]{biblatex}

addbibresource{biblatex-examples.bib}

DeclareInnerCiteDelims{footcite}{bibopenparen}{bibcloseparen}
DeclareInnerCiteDelims{cite}{bibopenparen}{bibcloseparen}

DeclareNameFormat{labelname}{%
  ifnumvalue{uniquename}<2
    usebibmacro{name:family-given}
      {namepartfamily}
      {namepartgiveni}
      {namepartprefix}
      {namepartsuffix}%
  else
    usebibmacro{name:family-given}
      {namepartfamily}
      {namepartgiven}
      {namepartprefix}
      {namepartsuffix}%
  fi
  usebibmacro{name:andothers}}

begin{document}
Cite with foonote and textcite.footnote{Cite[Vgl.][23]{nussbaum},
  and much more text.}
Cite with footcite.footcite[Vgl.][23]{nussbaum}
printbibliography
end{document}

I wouldn't use textcite here because you don't want its placement of pre- and postnote. But you can easily repurpose cite for that job.
In the example I would definitely prefer
 footcite[Vgl.][23]{nussbaum}

over
 footnote{Cite[Vgl.][23]{nussbaum}.}

but if you include additional text (as in the MWE), it may make sense to use the latter form.

footcite equals texcite with special format (parenthesis around year and given name initials)

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