MM - I think you are misunderstanding what the "next payment interval" means here. The next payment interval ends on 4/1/2016. Note that this IS in the next calendar year, which means you don't have to make a second payment by 12/31/2015.
The annuity payment interval cannot be longer than annual. So, you commence the RMD on 4/1/15, under an annual payment annuity payout. Let's just assume that the annual annuity payment will be $10,000 per year for life. April 1 of 2015, the recipient receives the first annual payment of $10,000. He does NOT have to receive a second payment by 12/31/15, because you are using the annuity payment method. HOWEVER, the next payment of $10,000 must be made April 1, 2016 (because the payment interval is annual.) It can NOT wait until 12/31/2016. The reason no one has been using 12/31/2016 is that this would fail to satisfy the minimum distribution requirement.
Hope this helps.