Amsbook class + two tables of contents

There are a few things to change in \starttoc to make it compatible with amsbook:

1. we want to avoid that it writes an entry in the table of contents, which \chapter* normally does in amsbook;
2. we don't want that it takes the initiative to set the headers, because amsbook takes care of the business anyway.

\documentclass{amsbook}
\usepackage{shorttoc}

\makeatletter
\renewcommand\shorttoc[2]{%
  \begingroup
  \c@tocdepth=#2\relax
  \@restonecolfalse
  \if@tightshtoc
    \parsep\z@
  \fi
  \if@twocolumn\@restonecoltrue\onecolumn\fi
  \renewcommand\@tocwriteb[3]{}%
  \chapter*{#1}%
  \@startshorttoc{toc}\if@restonecol\twocolumn\fi
  \endgroup
}
\makeatother

\title{Title}

\begin{document}
\maketitle

\shorttoc{Contents}{1}

\setcounter{tocdepth}{3}
\renewcommand*\contentsname{Contents (detailed)}
\tableofcontents

\chapter{Chapter}
\section{Section}
\subsection{Subsection}
\subsubsection{Subsubsection}

Some subsubsection text

\end{document}

The redefinition of \shorttoc is essentially the same as in the package, but

1. \@tocwriteb is (locally) disabled in order not to produce a TOC entry
2. \chapter*{#1} does not issue \@mkboth.

Short contents

Detailed contents

Below I define \firstToC to print the first Table of Contents and \secondToC to print the second one. The first one turns off overwriting of the .toc (setting \@fileswfalse, locally). The second removes the printing of the ToC within the ToC (again, locally).

\documentclass{amsbook}

\makeatletter
\newcommand{\firstToC}{{%
  \@fileswfalse% Don't overwrite the current ToC
  \@starttoc{toc}\contentsname
}}
\newcommand{\secondToC}{{%
  \renewcommand{\@tocwrite}[2]{}% Don't write ToC to ToC
  \renewcommand{\contentsname}{Contents (detailed)}%
  \@starttoc{toc}{\contentsname}%
}}
\makeatother

\begin{document}

\setcounter{tocdepth}{1}%
\firstToC

\setcounter{tocdepth}{3}%
\secondToC

\chapter{Chapter}
\section{Section}
\subsection{Subsection}
\subsubsection{Subsubsection}
Lorem ipsum\ldots

\end{document}