Time Zone support allows you to view and set events in a time zone other than the one in which you are currently located. The time in which you are viewing events can be set in the time-zone support preference.
Events created whilst time-zone support is off default to the native time-zone of the calendar which was set when it was created and turning time-zone support off lets the iPhone automatically shift the times of appointments to the local time.
A good explanation from apple of the time-zone support can be found here and here.
As for the design rationale, I believe that the confusing naming originates from the development of the feature in iCal. When the feature was built into iCal iCal displayed events in the time-zone of the computer clock which was set manually. Enabling time-zone support permitted "power users" to set the iCal time-zone independently of the computer clock.
If you want your events to act in the way you describe you must disable time-zone support and remember to enter the event times in the local time of the location in which you set it. Alternatively you must enable time-zone support, change to Eastern Time, enter the time of the event and alert in Eastern time, then disable time-zone support.
One thing to note is that time zone support can be simultaniously enabled on your computer and disabled on your phone. If your calendars are synced with iCal this allows you to use the computer to enter the event into the calendar and have it behave as you expect on your phone without mucking around in the settings all the time.
If you have a number of events you want to enter and you have their times in another zone, it's very handy to use Calendar's time zone support.
First, go to Calendar->Preferences, and click Turn on time zone support.
This will enable a dropdown in the titlebar of your Calendar window:
You can then use this menu to change the display of everything in your calendar to a new time zone. Pull it down and select Eastern time (or whatever other time zone or city your event is in), and Calendar will convert all your times to that zone.
At this point, you can start entering events, using the times listed in the program for the event. You will also see a new detail in the event detail popup:
This shows what time zone the event is in.
Once you've entered all the events, turn the time zone back. That will display all your events in your own time zone. If you are syncing with your phone (which knows what time zone it is in) all the events will remind you at the right time.
Best Answer
I had this problem. I drove 75 miles to nearest Apple store with an assigned appointment time just to have the "tech" flip a setting button from "Show" to "Hide".
In other words if you press "Hide" it hides the calendar, but in pressing "Hide" it changes the button to "Show". So when I was looking at the button which says "show", I assumed it was in show mode. Silly me. You have to press show to make it actually show, but then the button will display as "Hide", as if you're in hide mode, but you're not. It's an action button. Press for Show or Press for Hide. Whatever is displayed, your phone is doing opposite to that.
The button can be found by pressing Calendars to bring up a list of your calendars, then pressing the relevant Show button for the calendar. Alternatively, you can press the Show All Calendars button to unhide all the calendars at once.