After "Theorem", "Definition" the actual content should start at the same indent

Question

Since "Thm 1.2", "Def 1.33", etc. all have about the same width, I'd like to try a layout with a fixed indentation. The actual content after the label and number should begin at the same indendation if on the same line.

In the screenshot all text behind the labels should start at the red line. And ideally, also lists would left-align to this tab.

I tried with the tabu package but it seems tabulators are not exactly the use-case for a global layout scenario like this here.

How does one influence the layout to get this behaviour?

Edit: Here's a self-contained example:

\documentclass{article}

\usepackage[margin=1.25cm]{geometry}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{enumerate}
\usepackage{enumitem}

% amsthm theorem specifications
\theoremstyle{definition} 
\newtheorem{defn}{\sffamily Def}[section] 

\newtheorem{thm}[defn]{\sffamily Thm}
\newtheorem{lem}[defn]{\sffamily Lem}

% customize section headings
\usepackage{titlesec}
\titleformat*{\section}{\small\sffamily}

% abbreviations
\newcommand{\R} { \mathbb{R} }
\newcommand{\norm}[1] { \lVert #1 \rVert }

%%%%%%%%%%%%%%%
\begin{document}

\section{one}

\begin{defn}
$\hat{f}(\xi)=\int_\R f(x)e^{-ix\xi}dx$
\end{defn}

\begin{thm}
$|\hat{f}(\xi)| \leq \lVert{f}\rVert _1$ \\  $\hat{f} \in C(\R)$ and $\lim_{\xi \rightarrow \pm\infty}\hat{f}(\xi) = 0$
\end{thm}

{\defn    $(f \star g)(y) = \int_\R f(y-x)g(x)dx$
}

{\thm     $\norm{f \star g }_1 \leq \norm{f}_1 \norm{g}_1$
}
{\thm
\begin{enumerate}[label=(\roman*), font=\sffamily,labelindent=\parindent, leftmargin=*, align=left]    
\item $T_tf(x) = f(x-t)$
\item $M_\xi f(x) = e^{ix\xi}f(x)$
\item $D_a f(x) = a^{-1/2} f(x/a)$
\end{enumerate}
}

\end{document}

Answer 1

Update

In comments now it has been requested to indent not only the first line for each theorem-like structure, but to indent all lines; since internally amsthm uses a \trivlist to create the structures, the quickest way to achieve the desired result would be to use, foe example, etoolbox to patch some internal commands to change \trivlist to \list, thus allowing to define the desired \leftmargin value:

\documentclass{article}
\usepackage{etoolbox}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{enumitem}
\newlength\thmsep
\setlength\thmsep{5.5em}% change to suit your needs
\newtheoremstyle{fixedwidthindented}
  {\topsep}
  {\topsep}
  {\itshape}
  {0pt}
  {\bfseries\sffamily}
  {}
  {0em}
  {\llap{\makebox[\thmsep][l]{%
    \thmname{#1}~\thmnumber{#2}\if\relax\detokenize{#3}\relax.\fi}}%
    \thmnote{~{\normalfont(#3).}}}
\makeatletter
\patchcmd{@thm}{\trivlist}{\list{}{\leftmargin=\thmsep}}{}{}
\patchcmd{@endtheorem}{\endtrivlist}{\endlist}{}{}
\makeatother
\theoremstyle{fixedwidthindented}
\newtheorem{thm}{Thm}[section]
\newtheorem{defi}[thm]{Def}
\newlist{thmenumerate}{enumerate}{4}
\setlist[thmenumerate,1]{
  label=(\roman), 
  font=\normalfont\sffamily,
  labelindent=\parindent,
  leftmargin=,
  align=left,
}
\begin{document}
\section{Test Section}
\begin{thm}
A test theorem.
\end{thm}
\begin{defi}
A test definition with some filler text just to show how the text wraps when it spans more than one line with some filler text just to show how the text wraps when it spans more than one line.
\end{defi}
\begin{thm}
The following relations hold:

\begin{thmenumerate}
\item $T_{t} f(x) = f(x-t)$
\item $M_{\xi} f(x) = e^{ix\xi}f(x)$
\item $D_{a} f(x) = a^{-1/2} f(x/a)$
\end{thmenumerate}
\end{thm}
\begin{thm}
A test theorem with some filler text just to show how the text wraps when it spans more than one line 
[
M_{\xi} f(x) = e^{ix\xi}f(x)
]
with some filler text just to show how the text wraps when it spans more than one line.
\end{thm}
\end{document}

The result:

You can define a new style that uses a \makebox of the desired width to typeset the name and number for the structures; in the example below an eventual annotation will be typeset outside this box (this can also be changed if required, but the question didn't contain information about eventual annotations):

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsthm}
\newlength\thmsep
\setlength\thmsep{5.5em}% change to suit your needs
\newtheoremstyle{fixedwidth}
  {\topsep}
  {\topsep}
  {\itshape}
  {0pt}
  {\bfseries\sffamily}
  {}
  {0em}
  {\makebox[\thmsep][l]{%
    \thmname{#1}~\thmnumber{#2}\if\relax\detokenize{#3}\relax.\fi}%
    \thmnote{~{\normalfont(#3).}}}
\theoremstyle{fixedwidth}
\newtheorem{thm}{Thm}[section]
\newtheorem{defi}[thm]{Def}
\begin{document}
\section{Test Section}
\begin{thm}
A test theorem
\end{thm}
\begin{defi}
A test definition
\end{defi}
\begin{thm}
A test theorem
\end{thm}
\end{document}

Change the 5.5em in \thmsep according to your needs.

Update

In the case of a list beginning in the first of a theorem, the solution is a little more involved; the first item has to receive a "special treatment"; here's a little example:

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{enumitem}
\newlength\thmsep
\setlength\thmsep{5.5em}% change to suit your needs
\newtheoremstyle{fixedwidth}
  {\topsep}
  {\topsep}
  {\itshape}
  {0pt}
  {\bfseries\sffamily}
  {}
  {0em}
  {\makebox[\thmsep][l]{%
    \thmname{#1}~\thmnumber{#2}\if\relax\detokenize{#3}\relax.\fi}%
    \thmnote{~{\normalfont(#3).}}}
\theoremstyle{fixedwidth}
\newtheorem{thm}{Thm}[section]
\newtheorem{defi}[thm]{Def}
\newlist{thmenumerate}{enumerate}{4}
\setlist[thmenumerate,1]{
  label=(\roman*), 
  font=\normalfont\sffamily,
  labelindent=\parindent,
  leftmargin=\dimexpr\thmsep+\labelsep\relax, 
  align=left,
  start=2
}
\newcommand\FirstItem{%
\hangindent\leftmargini{\normalfont\sffamily(i)}\hskip\labelsep}
\begin{document}
\section{Test Section}
\begin{thm}
A test theorem
\end{thm}
\begin{defi}
A test definition
\end{defi}
\begin{thm}
\FirstItem  $T_{t} f(x) = f(x-t)$
\begin{thmenumerate}

\item $M_{\xi} f(x) = e^{ix\xi}f(x)$
\item $D_{a} f(x) = a^{-1/2} f(x/a)$
\end{thmenumerate}
\end{thm}
\begin{thm}
A test theorem with some filler text just to show how the text wraps when it spans more than one line
\end{thm}
\end{document}

Of course, the best solution here is to add some text before the list, which is a good practice anyways (adjust the left margin for the list to suit your needs):

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{enumitem}
\newlength\thmsep
\setlength\thmsep{5.5em}% change to suit your needs
\newtheoremstyle{fixedwidth}
  {\topsep}
  {\topsep}
  {\itshape}
  {0pt}
  {\bfseries\sffamily}
  {}
  {0em}
  {\makebox[\thmsep][l]{%
    \thmname{#1}~\thmnumber{#2}\if\relax\detokenize{#3}\relax.\fi}%
    \thmnote{~{\normalfont(#3).}}}
\theoremstyle{fixedwidth}
\newtheorem{thm}{Thm}[section]
\newtheorem{defi}[thm]{Def}
\newlist{thmenumerate}{enumerate}{4}
\setlist[thmenumerate,1]{
  label=(\roman*), 
  font=\normalfont\sffamily,
  labelindent=\parindent,
  leftmargin=\dimexpr\thmsep+\labelsep\relax, 
  align=left,
}
\begin{document}
\section{Test Section}
\begin{thm}
A test theorem
\end{thm}
\begin{defi}
A test definition
\end{defi}
\begin{thm}
The following relations hold:

\begin{thmenumerate}
\item $T_{t} f(x) = f(x-t)$
\item $M_{\xi} f(x) = e^{ix\xi}f(x)$
\item $D_{a} f(x) = a^{-1/2} f(x/a)$
\end{thmenumerate}
\end{thm}
\begin{thm}
A test theorem with some filler text just to show how the text wraps when it spans more than one line
\end{thm}
\end{document}