test: split browser tests into its own job, fix logs path

[avi-alpert]

Description

Split browser tests (Playwright) out of the e2e matrix job into a standalone browser-tests job so they run in parallel with the E2E suite.

Also fixes the failure debug step paths — test-results/ → browser-tests/test-results/ — so the PTY log and error contexts are actually captured on failure. Previously these pointed to the wrong directory, so failure diagnostics always showed (no pty log).

Related Issue

Closes #

Documentation PR

N/A

Type of Change

• Bug fix
• New feature
• Breaking change
• Documentation update
• Other (please describe): CI workflow improvement

Testing

• Verified the workflow YAML is valid
• browser-tests job has no needs: dependency, so it runs in parallel with the e2e matrix
• Browser tests don't use the CDK tarball or API keys from Secrets Manager, so those steps are correctly omitted
• Debug step paths now match Playwright's outputDir (browser-tests/test-results/)

Checklist

• I have read the CONTRIBUTING document
• I have added any necessary tests that prove my fix is effective or my feature works
• I have updated the documentation accordingly
• I have added an appropriate example to the documentation to outline the feature, or no new docs are needed
• My changes generate no new warnings
• Any dependent changes have been merged and published

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