Although eels are superficially similar to lampreys, in fact they are an entirely different creatures. In fact, as ray-finned fishes, eels are no more closely related to a jawless fish like the lamprey than is any other jawed animal (including humans).
In fact you are correct, however, everyone, I mean everyone that I know of in N. Calif. calls them eels, and always have. So take "eel" out of my statement and insert "lamprey" if it makes you feel better.