[Tech] Refactor how we write to game logs

[arielj]

We currently use explicit calls to writeFileSync and appendFileSync when we want to write game logs. This PR extracts that into more descriptive logger helper methods (initGameLog and appendGameLog) to simplify the usage so we can abstract the actual file name calculation and writing of the content.

I'm doing this because I want to add more logging and make it all async in the future, so I want to first simplify the API.

Nothing is changed in the functionality, logs should work just the same as before.

---

Use the following Checklist if you have changed something on the Backend or Frontend:

• Tested the feature and it's working on a current and clean install.
• Tested the main App features and they are still working on a current and clean install. (Login, Install, Play, Uninstall, Move games, etc.)
• Created / Updated Tests (If necessary)
• Created / Updated documentation (If necessary)

[flavioislima]

LGTM