You women can't decide what you want. A nanny calls out sick, you're pissed and would rather she let you make the call. She comes to work ill and she's gaming the system. She was willing to work OP and you sent her home. Your call, your dime.
It sounds like you didn't read the thread, or maybe you don't understand the OP's question.
The OP doesn't want the nanny working when she's sick. The problem is, nanny has used up all her PTO, on her choice to spend it all on a single trip. Thus, she has no sick time left. Nanny initiated the agreement and understood she would have no PTO left. Therefore, the OP is not responsible for paying her on the day she is too sick to work. Nanny gambled on her PTO needs, and lost.
Nanny made a very irresponsible choice when she planned the trip, and made a second irresponsible decision trying to come to work very sick. If I were OP, this would be a cause for concern about her judgment.