Allow time to reach the hole first
Rule 13 gives a reasonable time to reach an overhanging ball and then ten more seconds to see whether it falls. A fall during that allowance counts with the previous stroke. If it falls afterwards before being played, one penalty stroke is added.
Imagine a long putt stopping on the edge while the player is still several metres away. Starting a private ten-second timer at the instant of the stop omits the reasonable time to reach the hole. Conversely, standing beside it indefinitely does not preserve the allowance forever.
Count strokes separately from the dramatic fall
In an invented par-four example, the ball reaches the lip with the player’s third stroke. If it falls during the permitted wait, the hole is completed in three. If it falls after that wait before another stroke, the added penalty produces four. No extra visible putting stroke is necessary for those totals to differ.
The example is arithmetic applied to the rule, rather than a reported ruling. A television replay often compresses the walk and pause, so its edited duration may not establish which side of the waiting limit an actual incident occupied. Use the uninterrupted sequence and the official explanation when available.
Questions answered
Does the ten seconds always start when the ball stops?
A reasonable time to reach the hole comes first.
Can the score change without another visible stroke?
Yes. A late fall can add the specified penalty stroke.
Sources & further reading
Official references for the subjects discussed. Service terms and device software can change.