Yeah, except Midwife actually says you 'play' the card, while 'Flooded Vault' says 'put into play'. This comparison is hence faulty.
A: Yes, when you play a card you must follow all rules about how to play the card. Flooded Vault, much like Corrupted Midwife, only changes the location of the card it is letting you play the card from not how the card is payed for.
I'd asked the question, too, pointing out that difference, but didn't get an answer yet.