pgfplots scale legend position when below xlabel

Basically, what you are looking for is the key legend pos=outer south, which doesn't exist yet, as explained by the PGFPlots developer in the forum discussion you linked to. The reasons are that it's complicated to take into account whether there's an axis title below the axis or not, which has an influence on where to place the legend. However, it turns out that it's not actually that complicated: The following code snippet assigns a node alias to the x axis label, and then uses the code snippet from How can I know if a node is already defined? in the legend pos/outer south code to decide whether the axis title has to be taken into account or not:

\makeatletter
\pgfplotsset{
    every axis x label/.append style={
        alias=current axis xlabel
    },
    legend pos/outer south/.style={
        /pgfplots/legend style={
            at={%
                (%
                \@ifundefined{pgf@sh@ns@current axis xlabel}%
                {xticklabel cs:0.5}%
                {current axis xlabel.south}%
                )%
            },
            anchor=north
        }
    }
}
\makeatother

Full code:

\documentclass{article}

\usepackage{pgfplots}
\usepackage{amsmath}

\makeatletter
\pgfplotsset{
    every axis x label/.append style={
        alias=current axis xlabel
    },
    legend pos/outer south/.style={
        /pgfplots/legend style={
            at={%
                (%
                \@ifundefined{pgf@sh@ns@current axis xlabel}%
                {xticklabel cs:0.5}%
                {current axis xlabel.south}%
                )%
            },
            anchor=north
        }
    }
}
\makeatother

\newlength\figureheight
\newlength\figurewidth

\begin{document}

\setlength\figureheight{2.0in} 
\setlength\figurewidth{2.0in}
\begin{figure}\centering
    \begin{tikzpicture}[trim axis left]
        \begin{axis}[
        width=\figurewidth,
        height=\figureheight,
        xlabel={x label text},
        ylabel={y label text},clip=false,
        legend pos=outer south
        ]

        \addplot coordinates {(0,0) (1,1)};
        \addlegendentry{Legend Entry 1}

        \addplot coordinates {(0,1) (1,2)};
        \addlegendentry{Legend Entry 2}
        \end{axis}
    \end{tikzpicture}
    \caption{2" tall figure, x label covered} %
\end{figure}
%now with a larger figure, the xlabel will no longer be hidden
\setlength\figureheight{3.0in} 
\setlength\figurewidth{3.0in}
\begin{figure}\centering
    \begin{tikzpicture}[trim axis left]
        \begin{axis}[
        width=\figurewidth,
        height=\figureheight,
        xlabel={x label text},
        ylabel={y label text},
        legend pos=outer south
        ]

        \addplot coordinates {(0,0) (1,1)};
        \addlegendentry{Legend Entry 1}

        \addplot coordinates {(0,1) (1,2)};
        \addlegendentry{Legend Entry 2}

        \end{axis}
    \end{tikzpicture}
    \caption{3" tall figure, x label visible} %
\end{figure}

\end{document}

---

Alternatively you could place the legend relative to the xlabel node by first naming the node using xlabel style={name=xlabel} and then using legend style={ at={(xlabel.south)}, % Place legend relative to xlabel node yshift=-1ex, anchor=north }

That way, the legend will always be the same distance from the x axis label:

\documentclass{article}

\usepackage{pgfplots}
\usepackage{amsmath}

\newlength\figureheight
\newlength\figurewidth
\pgfplotsset{compat=newest}

\begin{document}

\setlength\figureheight{2.0in} 
\setlength\figurewidth{2.0in}
\begin{figure}
    \begin{tikzpicture}
        \begin{axis}[
        width=\figurewidth,
        height=\figureheight,
        xlabel={x label text},
        xlabel style={name=xlabel}, % Name the xlabel node
        ylabel={y label text},
        legend style={
                at={(xlabel.south)}, % Place legend relative to xlabel node
                yshift=-1ex,
                anchor=north,
                legend cell align=left
            } %
        ]

        \addplot coordinates {(0,0) (1,1)};
        \addlegendentry{Legend Entry 1}

        \addplot coordinates {(0,1) (1,2)};
        \addlegendentry{Legend Entry 2}

        \end{axis}
    \end{tikzpicture}
    \caption{2" tall figure, x label visible} %
\end{figure}
\setlength\figureheight{3.0in} 
\setlength\figurewidth{3.0in}
\begin{figure}
    \begin{tikzpicture}
        \begin{axis}[
        width=\figurewidth,
        height=\figureheight,
        xlabel={x label text},
        xlabel style={name=xlabel},
        ylabel={y label text},
        legend style={at={(xlabel.south)},yshift=-1ex, anchor=north,legend cell align=left} %
        ]

        \addplot coordinates {(0,0) (1,1)};
        \addlegendentry{Legend Entry 1}

        \addplot coordinates {(0,1) (1,2)};
        \addlegendentry{Legend Entry 2}

        \end{axis}
    \end{tikzpicture}
    \caption{3" tall figure, x label visible} %
\end{figure}

\end{document}