[38.0.0] minor: [JDK upgrade] Update forking task runner to use bin/run-java

[github-actions]

{
"source_pr": "19709",
"target_version": "38.0.0",
"message": "Backport #19709 to release branch for version 38.0.0",
"note": "Add the approved label to start the automatic backport process."
}