Using pgfplotspointgetcoordinates together with fillbetween command

Question

I'm curios with following strange behavior of my code (see MNWE below). If the command for filling area between function is added, than the coordinates A and B are changed. It is weird for me, because I am using the command pgfplotspointgetcoordinates for obtaining the correct coordinates for point-1 and point-2. What is going on?

The problematic code is:

  \addplot[fill=gray,opacity=0.4] 
    fill between[of=gx and fx,soft clip={domain=-1:1}];

And the figure is changed too much. The graph curves are changed and the coordinates for point A and B is touched.

I also see that the point coordinates are influenced by:

  enlarge y limits={rel=0.13},
  enlarge x limits={rel=0.07},

MNWE

\documentclass[11pt]{standalone}
\usepackage[usenames,x11names]{xcolor}
\usepackage{pgfplots}
  \pgfplotsset{compat=newest}
  \usepgfplotslibrary{fillbetween}
\usetikzlibrary{intersections}
\usepackage{amsmath}
\begin{document}
  \begin{tikzpicture}[
    thick,scale=0.5, 
    every node/.style={transform shape}
    ]
    \begin{axis}[
      domain = -2:2,
      restrict y to domain=0:4,
      grid = major,   % both
      grid style={line width=.2pt, draw=gray!20},
      major grid style={dashed, line width=.2pt, draw=gray!40},
      minor tick num=5,
      clip = true,
      clip mode=individual,
      axis x line = middle, 
      x axis line style={name path=xaxis}, % https://tex.stackexchange.com/questions/574555/tikz-refer-to-intersection-with-x-axis
      axis y line = middle,
      xlabel={(x)},
      ylabel={(y)},
      enlarge y limits={rel=0.13},
      enlarge x limits={rel=0.07},
    ]
  \addplot[
    name path global=fx,
    color=Gold3, samples=20, 
    smooth, ultra thick, 
    unbounded coords=jump, 
    no markers
  ]  gnuplot{2/(1+x^2)}; 

  \addplot[
    name path global=gx,
    color=Coral4, samples=20, 
    smooth, ultra thick, 
    unbounded coords=jump, 
    no markers
  ] gnuplot{x^2};  

  % fill area bounded by two curves
  % https://tex.stackexchange.com/questions/38461/
  \addplot[fill=gray,opacity=0.4] 
    fill between[of=gx and fx,soft clip={domain=-1:1}];

  \path [name intersections={of=fx and gx, name=point}];
  \node (prusecikA) [left,  font=\scriptsize] 
    at (point-1) {
    % -------------------------------------------------------------
    % using '\pgfplotspointgetcoordinates' stores the (axis)
    % coordinates of e.g. the coordinate (point-2) in
    % 'data point', which then can be called by '\pgfkeysvalueof'
      \pgfplotspointgetcoordinates{(point-1)}
      \(A=[
          \pgfmathprintnumber[fixed]{\pgfkeysvalueof{/data point/x}},
          \pgfmathprintnumber[fixed]{\pgfkeysvalueof{/data point/y}}
      ]\)
    };
  \node (prusecikB) [right,  font=\scriptsize] 
    at (point-2) {
      \pgfplotspointgetcoordinates{(point-2)}
      \(B=[
          \pgfmathprintnumber[fixed]{\pgfkeysvalueof{/data point/x}},
          \pgfmathprintnumber[fixed]{\pgfkeysvalueof{/data point/y}}
      ]\)
    };


  \draw[dashed] (-1,0) -- (point-1);
  \draw[dashed] (+1,0) -- (point-2);
  \node[below left] at (0,0) {\(0\)};
  \node[left, Gold3] at (1.5,2.3) {\(f(x) = \frac{2}{1-x^2}\)};
  \node[right, Coral4] at (-2,4) {\(g(x) = x^2\)};
  \fill [black] (point-1) circle (1mm);
  \fill [black] (point-2) circle (1mm);
\end{axis}

\end{tikzpicture}
\end{document}