Update the E2E Tests All workflow to run with Playwright

[eduardoumpierre]

Important

This issue depends on #10240. There is a PR open for the issue. Please build this issue's PR on top of PR #10276
Context threads:

• Original thread: p1738179003748179-slack-C01B8KNUYSW
• Follow up thread: p1738248569627589-slack-C01B8KNUYSW

Description

We need to either update the E2E Tests - All workflow or create a new one that matches the behavior of running the tests on the following setups:

• WP nightly and WC latest
• WP latest and WC 7.7.0
• WP latest and WC latest
• WP latest and WC beta

Acceptance criteria

• The "E2E Tests - All" workflow is updated (or a new one is created)
• The tests are passing
  • In case the tests are failing on a specific setup (a new issue can be filed to address that)

[eduardoumpierre]

I'm closing this issue since its related PR was already merged.