This will be difficult to explain and understand. I apologize.
Imagine you are flying from California to Tokyo. Now read this conversation:
Son: Are we in Tokyo yet?
Mom: No, we're still flying over the Pacific.
You are correct that both 'still' and 'yet' refer to 'at this moment'. However, 'still' implies that the condition has not changed. In my example, the airplane has been flying over the Pacific for several hours, so they are 'still flying'.
'yet', on the other hand, implies a future change in condition that has not happened. In my example, they have not arrived in Tokyo, so they are not 'in Tokyo yet'.