Indeed that Wikipedia definition misses the point.
Think in terms of a data flow diagram. Each processing node can have its output wired to 0 or more downstream nodes. In a visual diagram, those output wires look like a hand-held paper fan. Those wires or number of wires is its fan-out.
A node’s input can be wired to multiple sources. That’s its fan-in.
The term comes from electrical engineering of logic gates.