"Lode Runner II" is the title of a Commodore 64 game and an arcade machine. Graphically, they don't bear much resemblance to eachother so I doubt they are meant to be related (other than that they both bear the Lode Runner name and basic gameplay).
In the arcade there were a total of four Lode Runner machines (Lode Runner, Lode Runner II, Lode Runner III and a two player update to the original called Lode Runner V4).
On the C=64 there were two sequels (Lode Runner's Rescue and Championship Lode Runner).