Exact terminology of natural selection
"...in a single generation of species, the most fit individuals tend to survive.".  This is not correct. "Fitness" in an evolutionary context refers to how many offspring you produce. It has nothing to do with how many pushups you can do. 
Yes, individual organisms are selected, and the selection process involves differential survival (to reproduce).

"The process by which, over many generations, the genotypes that result in highest fitness tend to to be selected for. "

This is largely correct. 

"The "things that are being selected" are the genes..."  Yes and no. It is individuals that are selected. They live or die, depending on their traits, which are encoded by genes.

Darwin noticed the extreme physical variations that humans could create in dogs, and pigeons by selective breeding. He thought that the same process could occur in nature, and termed it "Natural selection".