TikZ does not accept intersection names other than `intersection-1` and `intersection-2`

Question

Here is my MWE:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\usetikzlibrary{intersections}
\begin{document}
\begin{figure}
\begin{tikzpicture}[scale=3]


\draw[name path={circle}] (0,0) circle(1cm);


\draw[name path={h1}] (-2,0) -- (2,0);
\draw[name path={h2}] (-2,0.8) -- (2,0.8);

    \fill[name intersections={of=circle and h2}]
    (intersection-1) circle(0.5pt)
    (intersection-2) circle(0.5pt);

\fill[red] (0,0) circle(1pt);

\draw[ultra thick] (intersection-1)--(intersection-2);

\end{tikzpicture}
\end{figure}
\end{document}

Simply, I am trying to mark the intersection points between a circle and a line, and draw a thick line between those points.

The above code works properly. But when I change the names of the intersection from (intersection-1) and (intersection-2) to something like i1 and i2, I get the error:

! Package pgf Error: No shape named i1 is known.
! Package pgf Error: No shape named i2 is known.

Must the names of the intersections begin with the word intersection?

Answer 1

https://tex.stackexchange.com/a/478227/197451

The relevant line is

\path [name intersections={of=D and E, by={C, C’}}];

Here the intersection points are computed and named C and C' ("name the intersection points of D and E by the names C and C'").

It is shortcut for

\coordinate (C) at ...;
\coordinate (C') at ...;

for some computed coordinates.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\usetikzlibrary{intersections}
\begin{document}
\begin{figure}
\begin{tikzpicture}[scale=3]


\draw[name path={circle}] (0,0) circle(1cm);


\draw[name path={h1}] (-2,0) -- (2,0);
\draw[name path={h2}] (-2,0.8) -- (2,0.8);

    \fill[red, name intersections={of=circle and h2, by={C, C'}}]
    (C) circle(1pt)
    (C') circle(1pt);

\fill[red] (0,0) circle(1pt);

\draw[ultra thick, red] (C)--(C');

\end{tikzpicture}
\end{figure}
\end{document}

edit

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\usetikzlibrary{intersections}
\begin{document}
\begin{figure}
\begin{tikzpicture}[scale=3]


\draw[name path={circle}] (0,0) circle(1cm);


\draw[name path={h1}] (-2,0) -- (2,0);
\draw[name path={h2}] (-2,0.8) -- (2,0.8);

    \path[name intersections={of=circle and h2, by={C, C'}}];

\fill[red] (0,0) circle(1pt);

\draw[ultra thick, red] (C)--(C');

\end{tikzpicture}
\end{figure}
\end{document}