is it possible to buy a mac and pay half when purchasing it then the other next month

You are arguing over nothing. the question was can he pay 1/2 down, 1/2 the next month. the answer is yes. I did it all without leaving the apple store website.

Well, not literally, no. In other words you, cannot purchase a Mac on layaway or something like that. No matter how you pay for it, Apple gets payment in full at the time of purchase.

If you use Apple's finance offer, that is actually through a private bank which has parterned with Apple. You will actually get a credit card from them, preloaded with the purchase amount already on its balance. Your options for paying that off are whatever the terms of the finance deal are (some interest free period, but you can pre-pay above the minimum payment amount as you wish).

Your payments can be whatever you want. If you want to pay it off the month after, you can do so, you are not locked in a monthly schedule.

Not if you purchase from Apple, no. You can arrange such with a third-party lender or credit card provider, but you pay Apple in full at time of purchase.

Not true, if you do the financing - available at the store or the website. I did that for my MBP thru the website, signed up for the financing, paid it off 2 months later.

Financing is through a third-party lender, NOT Apple. You arrange financing on your own, Apple gets paid in full.