That means that if a transcript physically showed up in AMCAS's mailbox on or before July 2, it should be marked off on their system as received. However, just because a transcript was marked as received doesn't mean they've started to verify anything.
Basically there are two queues - incoming transcripts, and transcripts to verify. When AMCAS gets transcripts in the mail, they go into the "incoming transcripts" queue and are marked off as received in order of the date received. Transcripts only enter the "transcripts to verify" queue once the applicant that submitted them has submitted their application. The date you see on the login page gives the status of the "incoming transcripts" queue and says nothing about the "transcripts to verify" queue.
The only use of that date is that if you got a transcript sent sufficiently long ago that it should have gotten from the school sending it to Washington, DC way before July 2, you might want to start thinking about whether the school forgot to mail it or it got lost in the mail.
