Draft: wip(web-ide): Use DefaultApiClient when authentication available

• Context MR: Draft: wip(web-ide): Use DefaultApiClient when ... (!1402 - closed)
• Part 1: refactor: Move DefaultApiClient out of GitLabSe... (!1418 - merged)
• Part 2: ...

Description

How has this been tested?

Screenshots (if appropriate)

Types of changes

• Refactor