After reading my blog on amethyst, in regard to "green amethyst" being a misnomer, I was asked "I've heard of a green sapphire, is that wrong too?". There's no straight answer to that. The term is certainly widely used and accepted, but the naming of gemstones is complicated.
Sapphire, by definition, is normally a blue stone, in fact that's what it's name means. It's the blue form of corundum, which like many gemstones, can certainly come in other colours, but when it does, we sometimes give them other specific names. If corundums are pinkish red we call them rubies, for example. Certainly, corundum can come in green, and other colours too, it all depends on impurities. Therefore a green sapphire is a rather confusing term to use but it's all we have (for now) and it's just one example of why arguments about names of gemstones are common, and can sometimes get a bit silly.
Many minerals, including corundum, beryl, quartz, and others, when pure, are actually colourless, but impurities such as iron, chromium, copper, and so on, cause coloured varieties. Colours can also be changed artifically in some minerals by heat, irradiation, or even dye. So if you use colour as a guide for the name, it could, theoretically, change name if it changed colour. Hmm.
It gets even more complicated on an international basis. When people think of jade, they usually think of a green stone. In fact a common name for jade in some countries is..."greenstone". But jade can come in many different colours. Then there's the issue of what precisely jade is - several different minerals are known as jade quite legitimately, and quite a few more simply "in the trade". In fact a definition of jade, if you want to be accurate, would necessarily take several paragraphs of explanation.
It's no wonder then that potentional buyers almost need a course in gemology. Then, popular names change with "trends" anyway, so you still have to keep up.
A popular gemstone recently is "mystic topaz". To be harsh, this is fake coloration. However, the process used is really not any more extreme than other common and accepted processes in the industry. Clear topaz is coated with a layer of titanium. You may be familiar with the rainbow effects from titanium, well, this is the rainbow being seen through a faceted clear stone. The effect is really quite something. When a customer in a jeweller's store asks if the stone is genuine, he will be told yes. It is, after all, real topaz. And a titanium coating is no different to the rhodium coating applied to many high end rings.
If instead he were to ask "is it pure?" he might get closer to the truth, but what is "pure"? As noted above, the colour of a gemstone is created by impurities.
A better question, if it matters to you, is to ask if the coloration is natural. There is a better chance of a straight answer then, but no guarantees. Salesmen can have creative ideas about what "natural" means too.
So why does a treated topaz get a fancy name like "mystic" while a naturally occuring gemstone like green sapphire gets nothing? In a word: marketing. Back when sapphires and rubies were first found, nobody knew they were the same mineral. At that point in time, if a green sapphire had been found, it would almost certainly have been called an emerald. And why not? Even today the word for emerald in some languages is the same word as that meaning "gemstone". At least when we say emerald, we are expecting green.