19 days after the close of the NEXT billing period.

From the example (regarding late fees):

Month 1; day 1: statement close; please pay by: Month 1, day 16.
Late fees are not assessed (i.e., you're not late) if you pay by the 19th day following the start of the NEXT billing period - i.e., Month 2, day 20.

So - charge something Jan 1, receive statement Feb 1, Pay March 19, all is good. I had always thought that payment had to be received by the end of the billing cycle - i.e., Bill from Jan 1, had to pay by Jan 31.

As to MR, with Platinum, you're supposed to be able to not pay until the 60 day mark and not forfeit MR points - i.e., in the above example, March 31. I do see that the verbiage from the link doesn't actually say that... digging through old stuff... it was true, if not I missed the change.