You can test for the proper heat by dropping a drop of water into the pan. The drop will sizzle and dance when the proper temperature is reached. Pour the eggs into the pan and work them from the edge to the center. Lift the edge and let the uncooked egg run under until it's all cooked. The topping or stuffing goes on top and the omelet is folded over and slid into a plate.
Poached eggs are just eggs cooked in water or stock.
The whole egg is dropped in the liquid and then fished out with a sieve or slotted spoon when it's done. This has to be done carefully to keep the egg from becoming ragged. Baked eggs are steamed in an oven in a ceramic container with a little cream and butter. Quiche is scrambled eggs and other things baked in a pie shell. Frittata is a quiche without the pie shell.