Switches the integration tests form Puppeteer/webdriver-manager to the
Bazel-managed Chromium/Chromedriver. This is now possible with the new
integration test rule for which we can consult the
`dev-infra/bazel/browsers` toolchain and setup environment variables.
This has been configured already in a previous commit.

This commit also includes some additional small cleanups necessary for
the new integration test rule:

* The `test.sh` scripts have been renamed as they would conflict with
  the `test.sh` scripts generated by the integration test rule.
  Previously this was not an issue because tests were declared at a
  higher-level. As mentioned though this has other downsides and it is
  trivial to rename the file.

* Related to the point above, since tests are now declared witin the
  actual test folder (for perf e.g.), `package.json` files setting
  `"type": "module"` will accidentally cause the `nodejs_test`-generated
  files to be considered ESM. This is not correct and likely needs to be
  fixed upstream in `rules_nodejs` where explicit `.cjs` extensions
  should be used. This is only happening **once** in the `injectable-def`
  test so it is acceptable doing that for now.

PR Close #44238