Refine Your Playgrounds overlay for autosaves

[adamziel]

What it does

Refines the Your Playgrounds overlay for the default autosave flow.

The overlay now separates opening existing Playgrounds from explicit creation
actions, and the E2E helpers/tests cover the autosave-first creation and restore
paths.

Rationale

The overlay is the highest-churn UI piece in the original save-by-default PR.
Keeping it in its own PR lets reviewers evaluate the creation flow and test
coverage without re-reviewing the persistence foundation.

Implementation

Updates the overlay component, supporting browser chrome styles, blueprint entry
points, and Playwright coverage for the default storage behavior.

This is stacked on feature/default-autosave-ux because the overlay presents
the autosave state and actions introduced there.

Testing instructions

npm exec nx run playground-website:typecheck
npm exec -- nx run playground-website:e2e:playwright -- --project=chromium --grep "Default Playground storage" --retries=0 --workers=1 --max-failures=1