I have a shared hosting in the hostgator and in the description of the plans vi that git client is accepted. I got in touch with Hostgator support to better understand details of the git client they are talking about, here's the answer:
"The difference between the Git server and Git Client, lies in functionality, where a Git client is just a user using external servers.
Since the Git server would be used to create a repository and provide other users, as it is not allowed to use shared servers as a repository, it is not possible to perform the procedure in P, M, Business, Plus, and Reseller plans. "
The question is, do I have a private repository on bitbucket, I can configure the server so that when pushing on my local machine, the changes are automatically saved in the repository and replicated to the site?