But in your case it would still mean a pull model imho. How would otherwise end up the mail in the Mendix app?
Even if you would create a mail address specific for this app it still means a pull from the app to this mail account to be able to process the mail. And I think that last option might be the best. Otherwise each user would need to give their credentials to the Mendix app to be able to retrieve the mail. And from a security perspective I would never allow this. With a special mail adress I can make a rule that certain mails need to be forwarded to this special mail address to be processed by the app. That gives me control.
I agree with Ronald, If you want to implement mail reading from Mendix it will be a PULL model.
Otherwise, if you are keen on PUSH model, then you must develop a consumable service from Mendix which can be called by an external application to PUSH the mails.
What that application is, will be up to you.
But if you are developing the external application too just for the sake of following the PUSH model, it is redundant layer.