The United States Postal Service (USPS) has implemented a new address verification policy, which can be done online, in person at a retail office, or by mail. The new policy requires customers to ...
I agree with three quarters of that. In BE we would use the verb "post" when putting the item into any kind of letter-box, public (Royal Mail) or private (the addressee's own), but I believe in AE the verb "mail" would only be used with a public mailbox (the US Postal Service) and not when they personally deliver a letter to someone's private mailbox.
Unfortunately, I have not been able to find a USPS Web site which states this clearly, and so for the moment will refer to this Web page by another source (which has links to USPS Web pages which are no longer active ).