You have a major reading comprehension problem.
There is no MobileMe account anymore and what you provided is no different - at all.
Copied from my post:
With an Exchange account - you can access a Gmail account and Hotmail account as an Exchange account, Apple iCloud account, and with a Yahoo account created with the Yahoo account preset which is not always reliable.
Exchange account is the same thing and is short for Microsoft Exchange. There is no MobileMe account anymore - Apple's iCloud account replaced that, and a Yahoo account created with the Yahoo account preset.
those that don't support push are POP, IMAP, AOL or Gmail.
Wrong.
An iCloud account is accessed as an IMAP account and the same for a Yahoo account that is created with the Yahoo account preset. An IMAP account can support push access which is up to the email account provider.
A Gmail account can be accessed as an Exchange account as included with my first post and the same for a Hotmail account.