CVS requires the ID, moneygram does not. Moneygram does not require the ID, but leaves it optional for the vendor to require an ID, according to Moneygram customer service. Technically, Moneygram is not requiring ID, but CVS does. I'm unable to get either Moneygram or CVS to commit to something in writing (email).
For what it is worth, I have emailed MoneyGram customer support, provided the details of the transaction, including their kiosk ID. I'll post something if/when I receive a response.