The two variations are a subset sort of thing. I just copied and pasted your answer and forgot to remove that. This will make it more clear.
If A is right (1/3 chance of that):
then the host reveals B
and switching does NOT pay off
OR
the host reveals C
and switching does NOT pay off
If B is right (1/3 chance of that):
then the host reveals C
and switching does pay off
If C is right (1/3 chance of that)
then the host reveals B
and switching does pay off
Now you're just being pedantic.
