I did a quick google on the topic and the only sensible thing was this Yahoo answer. However, I wanted to check here as well.
Also, according to Wikipedia corals are animals (marine invertebrates) but according to the Yahoo answer corals are skeletal remains of marine organisms, which way it is? If the latter, then presumably it would be fine for vegans to use, as it does not harm or exploit the organism that produced it, right?
I would say it's a matter of opinion whether it's OK for you to use the product, although from a strictly vegan perspective it might be considered probably not OK. But if the exact status of the coral in the product matters, then you will need to ask the manufacturer for clarity.
To elaborate, as Erica pointed out, both of your sources are correct; "coral" refers both to an animal and the skeletal remnant it produces. I think the "corals" in your shampoo are skeletal remains, but if you need to be sure of that, and know how it was harvested, you will have to ask the producer.
Vegans in general try to avoid using non-human animals or products derived from them. Strictly speaking, it does not matter which meaning of coral applies: coral is an animal product and is thus not vegan.
On the other hand, many people prefer to apply standards flexibly and consider using discarded animal remnants that are not a byproduct of any animal-exploiting practice to be perfectly ethical from a vegan perspective, and even beneficial since they may spare strain on the environment that would be caused by other ways of producing equivalent materials.
Additionally, since coral is endangered in many areas, and also since it provides habitats for other marine organisms (making it subject to the same concerns that cause some vegans to avoid palm oil, for instance), I would be wary of buying any product that contains it without very convincing reassurance that the coral was being obtained in a sustainable way.
I worked as a marine conservation educator for 15 years. Here's what I can contribute to this discussion. Unless you know for certain that coral rock was collected sustainably, i.e. already dead and collected on shore, don't buy, use, or exploit any product containing coral. Living coral is under tremendous environmental pressures caused by human activity, from pollution, global warming (the seas are profoundly affected, by rising temperatures and bleaching), commercial fishing, ocean bottom mining, and recreational indifference.
There are many types of coral, so appearance is not necessarily indicative of where it came from or how it was collected. Coral, as has been mentioned, can refer to the animal living in the exoskeletons they create, or the skeletal remains of their homes. Many reefs are made up of a combination of living and dead coral. Living coral often builds on the remains of old dead coral skeletons, which is why it should not be collected from the sea, where it forms the base of reefs and provides homes for many aquatic critters and plants. So, dead coral has a purpose. Even coral that's washed up into shores will eventually be crushed by the forces of the ocean into sand, though modest collection of washed up coral rock doesn't seem problematical in the scheme of things.
In some places, such as South Florida, there are deposits of ancient seabeds well inland that are quarried for coral rock to be used for building and for other purposes. That coral provides no benefit to living aquatic organisms, and in my opinion is not much different than any other quarried rock. Nothing lives on or in it.
So, if a product contains crushed quarried coral rock, I don't see any logical objection to its use.
As a sessile organism with behavior about as complex as a plant's, coral is arguably non-sentient, just like bivalves. If we define veganism as avoiding the use of products derived from sentient animals, then using coral would be vegan. (I'm not sure how ethical veganism would make sense if it cares about animals that can feel pain just as much as ones that don't feel anything.)
Of course, coral is an endangered species and is important for the marine ecosystem, so it might not be a good idea to use coral anyway, but those would be environmentalist reasons, not vegan reasons, strictly speaking.
