After downloading a fresh jenkins.war from the LTS URL http://mirrors.jenkins-ci.org/war-stable/latest/jenkins.war and launching it with a newly created JENKINS_HOME and correctly set environment variable, the update site points to the non-LTS version: http://updates.jenkins-ci.org/update-center.json. It should however point to http://updates.jenkins-ci.org/stable/update-center.json as documented at https://wiki.jenkins-ci.org/display/JENKINS/LTS+Release+Line.

This is probably not related to the use of JENKINS_HOME at all, because even removing all existing user-wide jenkins configuration doesn't cure this problem.