The way I understand Push is that everything about email, including status, is synched instantly.
Do you have a link where the way you understand it is stated?
I can't have email I read in the morning still showing as unread on the iPhone in late afternoon.
You can't huh?
When you select an IMAP account's Inbox mailbox or any server stored mailbox with the iPhone's mail client or with any email client, the mailbox is updated and synchronized with the server, which occurs in a very short period of time - in seconds after the mailbox is selected.
There is nothing in that definition that says what you said in your first post.
Nothing?
Sure there is.
Here is the definition.
+Push e-mail is used to describe e-mail systems that provide an "always-on" capability, *in which new e-mail is instantly and actively transferred (pushed) as it arrives by the mail delivery agent (MDA) (commonly called mail server) to the mail user agent (MUA), also called the e-mail client.* Most of today's clients are smartphones.+
And the following was copied from my first post.
+*Push access with email provides for when a new message is recieved at the incoming mail server, the incoming mail server "pushes" the message out to your iPhone instead of the iPhone's mail client having to check the incoming mail server for new messages* - via automatic fetching based on your settings for the automatic check, or manually checking.+
Your definition of "nothing" is certainly different from mine.