Sounds right to me regarding food. Then there is all the stuff to wear, toiletries, cleaning products, washing powder/liquid, make-up, bull's blood in cigarette filters, it's a bit of minefield beyond what we eat. Are there vegetarians who wear wool for instance? Vegans don't. Nor do they use honey in any way.
You didn't mention fish. Perhaps that's because you include fish as a type of meat. Anyway, to be clear vegetarians do not eat fish. If you eat fish but not (land) animal meat, then you are a pescatarian.
This varies and some people have an alternative usage and call themselves vegetarians when they eat fish but (strictly speaking) this is not correct.