I am a Pharmacist in New Zealand, [Home Pharmacy www.homepharmacy.co.nz. When we first consider the naming of the pharmacy we did consider Chemist [but after some google research - pharmacy is more popular than Chemist]
Chemist is an old term referring to the modern day Pharmacy. Mainly the older generations still use this term. Because back in the old days, pharmacists used to be called chemists - only because the used to make some medicines from scratch using chemistry. The term Chemist is mainly used in UK based and commonwealth countries like Australia, New Zealand, Canada etc.
When you went to the first shop to ask for the oil for your baby, you probably went to a Health Food shop that only sells natural supplements and food products.
So Chemist = Pharmacist / Pharmacy (but Chemist [solely deal with chemistry] as a profession is not the same as a pharmacist [deal with medicines]).