Integrate Multi-Environment Testing for Playwright (Web/App/In-Cluster)

[vyncent-t]

Dependent on issue #2332

Description

After migrating Playwright tests to the default structure, the next step is to enable running tests across different environments: web, app, and in-cluster. The goal is to create a unified setup so developers can easily run tests in any environment without needing reconfiguration, ensuring consistent test coverage across all modes.

[illume]

I think this can be done in another step. It's more important now to get at least one minimal app test running.

[vyncent-t]

Working on this #2344