I am assuming by "earth" you mean the British term for "ground" -- e.g. the ground wire that goes to a long rod stuck in the ground. In this picture "ground" is basically a "really big conductor" with an inexhaustible reservoir of charge (plus and minus) so that its potential never changes, no matter what electric field lines terminate on it. If this assumption is correct, here's your answer:
Yes, field lines pointing away from [toward] the surface of a conductor originate at [terminate on] that surface. This is because the charges in the conductor move under the influence of the fields until they reach a configuration on the surfaces of the conductor that exactly cancels the field inside the conductor. (Otherwise the charges would move some more inside the conductor until they did!) So your picture of field lines originating from positive charges and terminating on negative charges is intact; the necessary sources or sinks are just called up from the interior of the conductor to sit on the surface where needed.
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