FWIW, I believe it's "Ockanickon". It is located here, and also on the Southwest side of that lake across a bridge (Rainbow Bridge?):
https://online.boydsmaps.com/#16/39.8539/-74.7892/pines1999
I lived in various places in Medford from 1994 - 2006, and for awhile we owned one of the old log cabins in Medford Lakes while our daughter went to Shawnee. Used to ride bikes and wander around Camp Ockanickon.
This is Camp Lenape, which I'm not too familiar with, but my daughter had a good friend who lived in a development adjacent to it. My daughter thought it was so cool that her friend had that old theater "in her backyard"
https://online.boydsmaps.com/#16/39.8354/-74.8368/pines1999
Lots of great Medford memories. But honestly, I'm much happier down here in Atlantic County with a secluded little hideaway back in the woods that I'd never be able to afford in Medford, not to mention the all the traffic and development that has taken place up there.