chore(node-core): Fix node-core integration test assertions #19220

We excluded running the node-core integration tests from ci here, but we never ensured they were ran elsewhere.

This PR fixes the failing tests and adds a dedicated step in build.yml.

As for impact: The failing tests were mostly centered around asserting the correct sdk name, and two failing tests around usage of top-level await and one around adding headers to outgoing http requests.

None of these are critical, the last one working when scoped to node 22 (expected, since our http integration on node-core uses diagnostic channels that are only available on node 22). This is not an issue in our node sdk because we use otel's http integration for that.