Making the environment of A Cap B less thick

Question

How do I decrease the thickness of the line encompassing A intersection B? This is my Code:

\documeentclass{book}
\usepackage{tikz}
\usetikzlibrary{patterns}
\begin{document}
\begin{venndiagram2sets}[tikzoptions={scale=1.7, line width = 0.2cm}, labelA= {}, labelB={}, showframe=false, shade={}]
\setpostvennhook{
\draw (labelA) node[above=0.2cm, left=1.1cm]{\LARGE $A$};
\draw (labelB) node[above=0.2cm, right=1.1cm]{\LARGE $B$};}
\begin{scope}[pattern= horizontal lines]
\fillACapB
\end{scope}
\begin{scope}[pattern= north east lines]
\fillACapB
\end{scope}
\end{venndiagram2sets}
\end{document}

I want it to be something like this:

Answer 1

\documentclass[tikz, border=1cm]{standalone}
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}
\draw[line width=8pt] (0,0) circle[radius=2cm] (3,0) circle[radius=2cm];
\filldraw[fill=white] (0,0) circle[radius=2cm] (3,0) circle[radius=2cm];
\clip (0,0) circle[radius=2cm];
\fill[pattern=crosshatch] (3,0) circle[radius=2cm];
\end{tikzpicture}
\end{document}

\documentclass[tikz, border=1cm]{standalone}
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}
\draw[line width=8pt] (0,0) circle[radius=2cm] (3,0) circle[radius=2cm] (60:3) circle[radius=2cm];
\filldraw[fill=white] (0,0) circle[radius=2cm] (3,0) circle[radius=2cm] (60:3) circle[radius=2cm];
\clip (0,0) circle[radius=2cm];
\clip (60:3) circle[radius=2cm];
\fill[pattern=crosshatch] (3,0) circle[radius=2cm];
\end{tikzpicture}
\end{document}

Answer 2

Just to play a bit (if you want to do more complex diagrams, you'll have to calculate the angles more accurately).

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{intersections}

\begin{document}
    \begin{tikzpicture}
        \path[name path=circleA] (0,0) circle [radius=3];
        \path[name path=circleB] (4,0) circle [radius=3];
        \path[name intersections={of=circleA and circleB}] ;
        \draw[red,fill=orange] (intersection-1) arc(132:228:3) arc(-48:48:3);
        \draw[red,line width=3pt] (intersection-1) arc(48:312:3) arc(-132:132:3) -- cycle; 
    \end{tikzpicture}
\end{document}

EDIT

A second approach (which I don't really like because of the white fill which doesn't allow to have something under the picture, but anyway it's quite unlikely) is the following where the lines seem better to my eyes.

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{intersections}

\begin{document}
    \begin{tikzpicture}
        \path[name path=circleA] (0,0) circle [radius=3];
        \path[name path=circleB] (4,0) circle [radius=3];
        \path[name intersections={of=circleA and circleB}] ;
        
        \draw[red,line width=15pt] (intersection-1) arc(48:312:3) arc(-132:132:3) -- cycle; 
        \fill[white] (intersection-1) arc(48:312:3) arc(-132:132:3) -- cycle; 
        \draw[red,fill=orange] (intersection-1) arc(132:228:3) arc(-48:48:3);
    \end{tikzpicture}
\end{document}