Anesthesia is a state of being whereas you do not feel pain. An anesthetic is the agent that induces the state of anesthesia. The anesthesiologist will administer an anesthetic but you are under anesthesia. Examples: The dentist gave me a shot of Novocaine (an anesthetic) to induce local anesthesia before working on my teeth. I do not want general anesthesia for my surgery, I would prefer to receive a local anesthetic.

I'm not a doctor, but I'll try to explain. Anaesthesia is physical condition which is caused by special medicines, which are known as aesthetics. The correct term is "I'm under a local anaesthetic"

Good question. But to better understand the differences it is better to look at anesthesia images https://depositphotos.com/photos/anesthesia.html which will give you a visual understanding. And you will definitely understand the difference between the two.