The approach you've been using indeed does jar file with a string 'testing' in its name, as you specified, but the default install command sends it to your ~/.m2/repository directory, as seen in this output line:
/tmp/mvn_test/my-app/target/my-app-testing.jar to /home/maxim/.m2/repository/com/mycompany/app/my-app/1.0-SNAPSHOT/my-app-1.0-SNAPSHOT.jar
It seems to me that you're trying to generate a jar with such name and then copy it to a directory of your choice.
Try using outputDirectory property as described here: http://maven.apache.org/plugins/maven-jar-plugin/jar-mojo.html