Are the terms holistic health and integrative medicine the same thing?

Related not same. I see integrative as specifically alluding to the practice of using both conventional, western, modalities (aka allopathy) and traditional, often holistic healing systems, which are evidence based. Both aim to treat the whole person- body, mind, spirit- but holistic by definition does this by using a complete system or paradigm to diagnose, treat, prevent illness & maintain balance/health.
Should be . Integrative medicine is by definition holistic but has a more academic ring to it.
Holistic. No, holistic medicine tends to draw from complimentary and alternative medicine, while Integrative Medicine draws from the best evidence based treatments from traditional western medicine and complimentary and alternative medicine, treating the whole patient, mind, body and spirit.
Yes . Both terms describe an approach to medicine that tries to take the whole person into account - not just the medical diagnosis but how the person lives, eats , thinks, feels, and manages stress. Many people mistakenly use " holistic" to describe any medicine that isn't conventional, so it's been associated with a lot of fringe methods that are in no way holistic. "integrative" is less tarnished.