If I run 'git push' in terminal, it won't update my repository on github; however, if I initialize the git, and run the three following: 'git init', 'git add .', and finally, 'git commit -m "Initial commit"', then the repository updates.
Can someone help me?
Try doing it without running git init. That's the command t set up a new repository. Since you already have a clone there isn't anything to create.
Also, you need to actually have some changes to commit. Run a it status before running git add . to see if you actually have changes to commit.