Cutting to the chase, using a relative you pay will be questioned. And if I were the other side's lawyer I would require that you send a 1099 to the IRS with my client being copied, and I would have him contact the IRS to ensure it was actually reported, to ensure that the person you paid, pays income taxes on it (and helps ensure that there is nothing funny going on here). But the bottom line is what is acceptable is what the court allows in your situation.
This should not be considered legal advice and is intended for educational purposes only. It does not constitute a contract for legal services between any parties. Answers are given to questions for which there may be additional facts not mentioned which might change the legal issues or consequences.
In my experience work-related child care can be any of the suggestions you made. It truly depends on what you and your ex agree to, and if there is no agreement, what a court finds to be reasonable and in the best interests of the child. It would be helpful in terms of having proof that your sitter gives you an invoice for the hours and you pay the sitter by way of check. There is also a website called care.com where you can actually pay the sitter through the site and it will keep records of hours and how much is paid.
I agree with Mr. Popovich. If you are using a non-relative to provide extended child care in order to enable yourself to work, it will likely be accepted as legitimate - especially if you document those expenses with photocopies of checks.