Short answer: The pan is too hot. That explains why the outside is overcooked or burnt and the batter on the inside is undercooked. Lower your heat. Your batter comes into the equation too. If you have thick batter the inside will take longer to cook. So you need less heat so the outside doesn't burn but the inside batter gets completely cooked. Because you're using raw eggs, you need to make sure the pancakes are cooked through. Although remote, there is a chance of salmonella poisoning. Salmonella and eggs is a whole another topic by itself.
I put a little cooking oil in the batter but otherwise I don't grease the griddle. The browning is more even (attractive). Keep in mind I am using non-stick. Generally speaking, I don't like non-stick but I do keep a couple pans around for eggs, omelets and... pancakes.
Medium heat is correct. I have a non-stick griddle that covers two burners on the stove. I heat it until little droplets of water "dance" and the surface and quickly evaporate. Just stick your hand under the faucet and flick a few droplets onto the griddle. That's when my first pancakes hit the griddle. After the batter bubbles appear and the edges start looking dry, I will lift the edge to check for the proper brown-ness. For your subsequent batches, you may have to adjust the heat up or down or adjust how long you leave the cakes on the griddle.
If you want to be a little more scientific, you may want to invest in an infra-red thermometer. That will tell you what the temp of the griddle surface is. Most non-sticks should not be heated to more than 450-500 degrees. If it's a PTFE surface i.e. Teflon, overheating may give off toxic gases. Those gases will kill pet birds. If they can kill birds, they probably aren't good for you.
All things considered, lower your heat. When you have the right temp, your batches should come out perfect and repeatable. There is a certain amount of "knowing what's right" that comes with experience.