My apologies to Zelig. Anyway:
Safety:
Google – The condition of being protected from or unlikely to cause danger, risk, or injury.
Wiktionary – The condition or feeling of being safe; security; certainty.
Health:
Google – The state of being free from illness or injury.
Wiktionary – The state of being free from physical or psychological disease, illness, or malfunction; wellness.
So I’d have to agree with Lior:
“protecting one’s health is […] a matter of safety.”
So both of you are right and wrong. You are right that health is not the same as safety, but Zelig is right that wearing a coat is safety.