IT’S and ITS

A while back, I've had an argument with a British friend that said no-one used it's anymore, everyone writes its. I couldn't believe that to be true, it seemed impossible, but I let it go.

I've just watched a Cheers episode from 1985 and I couldn't believe my eyes. Take a look at the two screenshots below and tell me what you think.

How come they printed that? Is it not a mistake? Yes, it is! It should be its in both places! And why?

Its is the possessive pronoun; it modifies a noun.
It's is a contraction of it is or it has.

Incorrect: The mother cat carried it's kitten in it's mouth. (Possessive pronoun, no apostrophe)

Correct: The mother cat carried its kitten in its mouth.

Correct: I think it's going to rain today. (Contraction of it is)

Correct: It's been a very long time. (Contraction of it has)

Isn't this basic grammar?