Are coral reefs a Biochemical Sedimentary rock?
Yes, limestone indeed is "biochemical".

The main categories of non biochemical sediments are sandstone, mudstone, shale, conglomerate,.. if you put the Sahara for example under lithification it will become sandstone, since it consists mostly of quartz.. almost all continental sedimentary rocks are "non-chemical" they just vary by the components, whilst most marine sediments are biochemical, as you say..they consist mostly of carbonate and aragonite skeletons (shells in most cases) of dead animals.. that's what makes up limestone.. the main corpus of coral reef is a good example for mostly pure limestone.. The carbonate classification is based on the amount (and form) of the skeletons.. You should have a look at it, if that's interesting to you.
Coal, as you mentioned, would be another biochemical sediment but it's not only made of skeletons but the pure organic matter of dead animals, but it's genesis is rather complicated..

So to answer your question: Yes, coral reefs are biochemical sediments and very important ones, too, since they grow really fast! 
As noted in your quotation: Biochemical sedimentary rocks are created when organisms use materials dissolved in air or water to build their tissue!
That's exactly what the mentioned polyps do.. You can also have a look at stromatolithes build by cyanobacteria.. that's a nice and understandable example to understand how that works.