Nodes in PGFPlots Get Cut Off

I’m trying to draw a basic graph in pgfplots but facing an issue that I wasn’t able to solve based on similar posts. I’m trying to add the functions I’m displaying on my graph on their respective lines using nodes, but I end up getting the following result:

usepackage{pgfplots}

usepgfplotslibrary{fillbetween}

begin{tikzpicture}
  begin{axis}[axis lines=center,
    xlabel=$x$,
    ylabel=$y$,
    xmin=-1,
    xmax=1.5,
    ymin=-1,
    ymax=2.5,
    xtick distance=1,
    clip=true,]
    addplot[name path=A, thick, black, samples=200]{x} node[pos=0.61, right]{$y=x$};
    addplot[name path=B, thick, black, samples=200]{x + 1} node[pos=0.61, right]{$y=x + 1$};
    addplot[fill=red]fill between[of=A and B, soft clip={domain=0:1}];
  end{axis}
end{tikzpicture}

As you can see from the image, y = x _+ 1 gets cut off, while y = x is just at the edge of it, barely saving it. I tried setting clip=false but my graph goes haywire, getting displaced from the center and going all over my page:

Is there a way to fix this?

usepackage{pgfplots}
usepgfplotslibrary{fillbetween}

begin{tikzpicture}
  begin{axis}[axis lines=center,
    xlabel=$x$,
    ylabel=$y$,
    xmin=-1,
    xmax=2,
    ymin=-1,
    ymax=2.5,
    xtick distance=1]
    addplot[name path=A, thick, black, samples=20, domain=-1:2]{x} node[pos=0.61, right]{$y=x$};
    addplot[name path=B, thick, black, samples=20, domain=-1:2]{x + 1} node[pos=0.61, right]{$y=x + 1$};
    addplot[fill=red]fill between[of=A and B, soft clip={domain=0:1}];
  end{axis}
end{tikzpicture}