People seem to confuse pocket healer with support I've noticed. Pocket healer is a KIND of support, but lulu definitely doesn't have the pocket healer kit.... seeing as she has no heal.
Defensive support: Focuses on keeping the ally alive through heals, shields, and debuffs.
Offensive support: Focuses on keeping the ally alive by keeping the enemy away from the ally, by being able to harass and disrupt the enemy's playstyle.
Lulu obviously fits under the offensive support category. And I love it.
I just wish I could buy her. She costs so much right now. QQ