How does one NOT save Messages history upon new login in El Capitan on iMac? "save history when conversations are closed"

That is a preferences option in Message app on El Capitan. I do NOT have this box checked, yet every time I log into Messages EVERY conversation from at least the last week, if not longer, populates the app. How do I get this to stop happening and start fresh each new login session? And is it possible to have the Messages app synced with the iOS Messages for when I delete a conversation - having it deleted on both?

Thanks!
The pref doesn't say "Save history when you quit or logout" it says "Save history when conversations are closed" - i.e., when you hit the X to get rid of them - either permanently, or temporarily, depending on that pref.

If you want to delete the archive itself, you'd have to do that manually.
Files live in ~/Library/Containers/com.apple.iChat/Data/Library/Messages