How to deploy to multiple servers?

Question

Software Deployment

Hey!
Please tell me the names of the tools in the following situation:
- there is a certain service (a set of php scripts) that is hosted on github.
- sometimes there are changes and you have to go to 5-6 servers, do a git pull.
- then correct the changes in the config script with handles, tk. git contains only config.example, and on servers it already has server-specific settings, and of course, such a file is not controlled by git.
I want something like the following
- A git push to master happened
- I prepared a patch for the config files (some setting was added) and clicked the "Apply" button
- A git pull happened on the serverAX, my patch rolled
- Ideally, the frontend twitched for errors
- I saw the report
The algorithm is quite simple, and it can be implemented in 10 different ways and means, but I don’t want to reinvent the wheel.
By what automatic means do you roll out code changes to the servers?