When did L switch the 2nd Death Note in "The Last Name"?

Question

In "Death Note: The Last Name", L Reveals that the Death Note Misa had which Light used to try and kill his father was a fake, however after L's fake death Light touched the Death Note to see Ryuk again.

I am wondering when L switched the Death Notes?

Answer 1

The facts are

1. At 1:56:10-20 Light can clearly see Ryuk, which means he should have touched the second DN at least once.
2. At 1:56:30 Light writes down his father's name, who doesn't die in the end.

In those 10 seconds it would have been impossible for L to switch the notebooks. Assuming this isn't just a mistake from the authors, I can think of two possibilities

1. Either L had secretly touched Light with a piece of the Death Note and asked Ryuk to play along and to only appear from the moment light had touched the fake notebook, making Light believe that it was the real one.
2. They only switched the pages and not the cover, as they did in the manga.

My gut feeling says it's the second, but I can't really prove it.