Typeset saturation function

Using a technique of egreg at Define macro that scans ahead for super- and/or subscript, "absorbs" them and moves their argument, I let xparse help with redefining ^ and _ to handle the kerning.

This MWE uses the saturation.pdf supplied by the OP's code.

\documentclass{article}
\usepackage{xparse,mathtools}

\NewDocumentCommand{\sat}{e{^_}}{%
  \mathop{\IfValueT{#1}{\phantom{\scriptscriptstyle#1}\mathchoice
   {\kern-5pt}{\kern-5pt}{\kern-3pt}{\kern-2pt}}%
   \rawsat\mathchoice
   {^{\IfValueT{#1}{\mathllap{\scriptscriptstyle#1\mkern14mu}}}%
    _{\IfValueT{#2}{\scriptscriptstyle\mkern-13mu#2}}}%
   {^{\IfValueT{#1}{\mathllap{\scriptscriptstyle#1\mkern14mu\strut}}}%
    _{\IfValueT{#2}{\scriptscriptstyle\mkern-13mu#2}}}%
   {^{\IfValueT{#1}{\mathllap{\scriptscriptstyle#1\mkern11mu\mathstrut}}}%
    _{\IfValueT{#2}{\scriptscriptstyle\mkern-9mu#2}}}%
   {^{\IfValueT{#1}{\mathllap{\scriptscriptstyle#1\mkern9mu}}}%
    _{\IfValueT{#2}{\scriptscriptstyle\mkern-6mu#2}}}%
}}
\usepackage{scalerel,graphicx}
\newcommand\rawsat{\ThisStyle{\kern.5\LMpt
  \includegraphics[height=2\LMex]{saturation}\kern.5\LMpt}}

\begin{document}
$x=\sat_{-5}^{10} u(t)$

$\scriptstyle x=\sat_{-5}^{10} u(t)$

$\scriptscriptstyle x= \sat_{-5}^{10} u(t)$
\end{document}

And just to prove I have resolved the kerning even as the superscript width changes, here is the same example with a wider superscript:

ADDENDUM:

Here is a version that does not require an external image file to be dragged around, but instead saves the tikz image of it at initialization.

\documentclass{article}
\usepackage{xparse,mathtools,stackengine,tikz,scalerel}
\NewDocumentCommand{\sat}{e{^_}}{%
  \mathop{\IfValueT{#1}{\phantom{\scriptscriptstyle#1}\mathchoice
   {\kern-5pt}{\kern-5pt}{\kern-3pt}{\kern-2pt}}%
   \rawsat\mathchoice
   {^{\IfValueT{#1}{\mathllap{\scriptscriptstyle#1\mkern15mu}}}%
    _{\IfValueT{#2}{\scriptscriptstyle\mkern-14mu#2}}}%
   {^{\IfValueT{#1}{\mathllap{\scriptscriptstyle#1\mkern15mu\strut}}}%
    _{\IfValueT{#2}{\scriptscriptstyle\mkern-14mu#2}}}%
   {^{\IfValueT{#1}{\mathllap{\scriptscriptstyle#1\mkern12mu\mathstrut}}}%
    _{\IfValueT{#2}{\scriptscriptstyle\mkern-10mu#2}}}%
   {^{\IfValueT{#1}{\mathllap{\scriptscriptstyle#1\mkern10mu}}}%
    _{\IfValueT{#2}{\scriptscriptstyle\mkern-7mu#2}}}%
}}
\newcommand\rawsat{\ThisStyle{\kern.5\LMpt
  \scaleto{\rawsatimg}{2\LMex}\kern.5\LMpt}}
\savestack\rawsatimg{%
\begin{tikzpicture}
\coordinate (A) at (-.7,-.6);
\coordinate (B) at (-.2,-.6);
\coordinate (C) at (+.2,+.6);
\coordinate (D) at (+.7,+.6);
\draw [black, line width=3pt] (A)--(B)--(C)--(D);
\end{tikzpicture}%
}
\begin{document}
$x=\sat_{-5}^{10} u(t)$

$\scriptstyle x=\sat_{-5}^{10} u(t)$

$\scriptscriptstyle x= \sat_{-5}^{10} u(t)$
\end{document}

A solution with tikz.

\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage{xparse}

\makeatletter
\newif\ifsat@limits
\sbox\z@{$\sum$}
\sbox\@ne{$\displaystyle\sum$}
\tikzset{sat/.search also=/tikz}
\def\satset#1{\tikzset{sat/.cd,#1}}
\satset{
  height/.store in=\sat@ht,
  is limits/.is if=sat@limits,
  nodisplay/.code={
    \@tempdima=\dimexpr\dp\z@+\ht\z@
    \pgfkeysalso{height=\@tempdima}
  },
  display/.code={
    \@tempdimb=\dimexpr\dp\@ne+\ht\@ne
    \pgfkeysalso{height=\@tempdimb}
  },
  nolimits/.style={is limits=false},
  limits/.style={is limits=true},
}
\NewDocumentCommand {\satpic} { O{} m m m } {
\begin{tikzpicture}[baseline=0.5\dp\[email protected]\ht\z@, x=1ex, y=1ex, sat/.cd, nolimits, nodisplay, #1]
  \coordinate (A) at (-\sat@ht/2,-\sat@ht/2);
  \coordinate (B) at (-\sat@ht/7,-\sat@ht/2);
  \coordinate (C) at (\sat@ht/7,\sat@ht/2);
  \coordinate (D) at (\sat@ht/2,\sat@ht/2);
  \draw [black, line width=.6pt, #2] (A)--(B)--(C)--(D);
  \ifsat@limits
    \IfValueT {#3} {
      \node [below=1.3pt, inner sep=0pt] at (current bounding box.south) {$\scriptstyle #3$};
    }
    \IfValueT {#4} {
      \node [above=1.3pt, inner sep=0pt] at (current bounding box.north) {$\scriptstyle #4$};
    }
  \else
    \IfValueT {#3} {
      \node [anchor=west, inner sep=0pt, xshift=.3ex] at (B) {$\scriptstyle #3$};
    }
    \IfValueT {#4} {
      \node [anchor=east, inner sep=0pt, xshift=-.3ex] at (C) {$\scriptstyle #4$};
    }
  \fi
\end{tikzpicture}
}
\makeatother
\NewDocumentCommand {\sat} { E{!@_^}{{}{}} } {
  \mathchoice
  {\satpic[display, limits, #1]{#2}{#3}{#4}}
  {\satpic[nodisplay, nolimits, #1]{#2}{#3}{#4}}
  {\satpic[nodisplay, nolimits, #1]{#2}{#3}{#4}}
  {\satpic[nodisplay, nolimits, #1]{#2}{#3}{#4}}
}

\begin{document}
Saturation A: $\sat_{-5}^{10}$ and $\sat!{limits}_{123}^{456}$

Saturation B:
\[
  \sat@{red}_{10}^{200} \sat@{line width=1pt}_{40000}
\]

\end{document}