I'm having basically the same exact problem as sbaker was having as was discussed here: https://premium.wpmudev.org/forums/topic/membership-paypal-emails
I get those exact same three emails.
Sbaker resolved his issue by using only one payment gateway. But I only have one payment gateway activated to start with (Paypal Payments Standard).
I plan on having members move through a succession of finite access levels, one per month. Am I using the wrong payment gateway?