He's vague about induction, and as ong pointed out 1/c is not a time, so his conclusion there is, without further explanation from him, nonsense.
I suspect it's an accurate use of the term. He said 1/c, not 1/c seconds. The time is already factored into c, it's just not obvious.

If it takes 1/c, that's the same as saying it takes 1/300000000m/s - the "per second" is there. But it has distance, too. So it takes 1 second for every 300000000m in distance... the speed of light. If the distance between switch and bulb is 1 meter, then it takes 1/300000000 of a second in time.

1/c simply seems like the correct way to write down the speed of light as a function of time over distance, rather than distance over time.