I've read through most of this thread and spoken with Apple. They have accepted screenshots and said they are working on it. It is clear the problem is Apple/iOS 8.
But here is a question for those who have a more technical bent - I have found that on an even-by-event basis, the following will correct an entry. It is far too complicated to be a reasonable workaround, but it raises the question, "why does this stop the problem?" considering the breadth to the issue as described by a number of commenters here. In other words, will it help someone find an answer to what is causing it if the following stops it:
1) Enter an event as you normally would. 2) Select "Add". 3) GMT shows up. 4) Go back and edit the event (notice that the start time now shows GMT in the edit mode) 5) select "starts" and when the start-time wheel shows up, select "Time Zone" and reset it to the local time zone (this requires clearing the GMT and entering a few letters of the nearest large city in your time zone). 5) be sure to set the time again to the correct beginning time (because OS has also now set the starting time to the GMT equivalent). 6) Select "Done" 7) Return to the date display.
What is interesting (once you get over the basic issue, which is maddening) is that now the GMT is gone. And you can drag it to a different start time and the GMT will not reappear. If you go through those steps in the original setting of the appointment, iOS won't get it right. So why does it get it right when you enter the same information the second time during the edit.
As I said, it is way too cumbersome to do this for every appointment, but why does essentially the same setting the second time convince iOS that you just want your local time but it doesn't make it work initially?