In Gmail, if I click on "Inbox", I see all the messages in my Inbox, but none show the "inbox" label. If I search for "in:inbox", I see the same messages, but now each shows the "inbox" label. If I click on "All Mail", the messages in my inbox are included in the results, and each has an "inbox" label. If I click on a message (regardless of whether it's in my Inbox) and select the Labels drop-down menu, there's no "inbox" label for me to choose. Why is the "inbox" label only sometimes visible?
I find the "read" label especially confusing. Searching for "label:read" shows all my read mail, but none of the messages has a "read" label on it. There's no "Read" label listed in the left-hand navigation area, but if I try to create a label named "Read", I'm told "Sorry, you can't create a label named "read" (it's a reserved system label)." As is the case with "inbox", the Labels dropdown doesn't offer a "read" option. What's the story with the seemingly-invisible "read" label?