ci: Update TypeScript test workflow for improved Playwright integration (#4781)

* Update TypeScript test workflow to improve Playwright caching and containerization

- Add container image for Playwright to ensure consistent environment
- Modify Playwright version retrieval to use `npm ls` for accuracy
- Enhance caching strategy with restore keys for Playwright binaries
- Refactor Playwright installation steps to handle dependencies based on cache status

* Remove Playwright container image specification from GitHub Actions workflow

* Add GitHub Action to install Playwright with caching support

- Created a new composite GitHub Action `install-playwright` to install Playwright and its dependencies with caching.
- Updated `typescript_test.yml` workflow to use the new `install-playwright` action, simplifying the installation process and ensuring efficient use of cache.
- The action supports specifying a working directory and selecting browsers to install.

* Remove redundant Playwright caching steps from GitHub Actions workflow

* Update Playwright version extraction logic in GitHub Action

- Modify script to read `package.json` directly for Playwright version.
- Support both `dependencies` and `devDependencies` for version retrieval.
- Remove caret and tilde symbols from version string.

* Optimize Node.js setup and caching in GitHub Actions workflow

* Set UV_CACHE_DIR environment variable in GitHub workflows

* Refactor SignUpPage to import InputComponent from the core components directory

* Set default test suites to an empty array in TypeScript test workflow

* Fix conditional logic in TypeScript test workflow for release builds

* Fix syntax error in conditional statement in GitHub Actions workflow