Start-By Enviorment Varialbe plugin works olny with "Force Promotion" of Promoted Builds plugin.

In a promotion process, if option "Only when manually approved" (under section "Criteria") is selected, and the user promotes the build (on "Promotion Status" page) by clicking "Approve" button, JENKINS_STARTED_BY will not be correctly populated. In console output:

Legacy code started this job.  No cause information is available
...
+ echo JENKINS_STARTED_BY=
JENKINS_STARTED_BY=

If this option is not selected, therefore the user can only promote a build by clicking "Force Promotion" (if he has the privilege), then JENKINS_STARTED_BY variable is populated correctly. In console output:

Started by user john
...
+ echo JENKINS_STARTED_BY=john
JENKINS_STARTED_BY=john

In attachment are screen-shots of a trivial Jenkins job configuration demonstrating the test cases.