A subset of the intersection in a Venn Diagram with tikz

Question

I would like to delete, from the following figure:

the top circle, but without removing the black edge that separate the orange part from the gray part. The figure above has been obtained by the following code:

\documentclass[border=0.2cm]{standalone}
% Required packages
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[thick,
set/.style = {circle,
     minimum size = 3cm,
     fill=black!30}]
% Set A
\node[set,label={135:$A$}] (A) at (0,0) {};
% Set B
\node[set,label={45:$B$}] (B) at (1.8,0) {};
% Intersection
\begin{scope}
\clip (0,0) circle(1.5cm);
\clip (1.8,0) circle(1.5cm);
\clip (0.9,1.5) circle(1.5cm);
\fillorange!60 circle(1.5cm);
\end{scope}
% Circles outline
\draw (0,0) circle(1.5cm);
\draw (1.8,0) circle(1.5cm);
\draw (0.9,1.5) circle(1.5cm);
\end{tikzpicture}
\end{document}

A rough example of the desired result is:

Answer 1

From your code with minor changes: using \draw[fill=orange!60] (0.9,1.5) circle(1.5); and removing \clip (0.9,1.5) circle(1.5cm);

\documentclass[tikz,border=0.2cm]{standalone}
\begin{document}
\begin{tikzpicture}[thick]
\draw[fill=black!30] 
(0,0) circle(1.5) +(135:1.8) node{$A$} 
(1.8,0) circle(1.5) +(45:1.8) node{$B$};        
% Intersection
\begin{scope}
\clip (0,0) circle(1.5);
\clip (1.8,0) circle(1.5);
\draw[fill=orange!60] (0.9,1.5) circle(1.5);
\end{scope}
% Circles outline
\draw (0,0) circle(1.5cm);
\draw (1.8,0) circle(1.5cm);
\end{tikzpicture}
\end{document}

Update Here is what I recommend: 1. make the code more flexible, easier to control, use variables \r as radius of two circles, and \d as distance from two centers (to the origin); 2. make the intersection as Reuleaux triangle (look better ^^).

\documentclass[tikz,border=0.2cm]{standalone}
\begin{document}
\begin{tikzpicture}[thick]
\def\r{1.5} % radius of two circles
\def\d{1}   % distance of two centers
\def\cirA{(210:\d) circle(\r)}
\def\cirB{(-30:\d) circle(\r)}
\draw[fill=violet!30] 
\cirA +(135:\r+.3) node{$A$} 
\cirB +(45:\r+.3)  node{$B$};
% Fill intersection
\begin{scope}
\clip \cirA;
\clip \cirB;
\draw[fill=yellow] (90:\d) circle(\r);
\end{scope}
% Circles outline
\draw \cirA \cirB;
\end{tikzpicture}
\end{document}