The confusion stems from the word organic and inorganic. The defined meaning of organic is something which derives from a carbon source, whilst inorganic means from a non-carbon source, like metals, copper ions, iron, etc. All living things need carbon, and cannot grow without it, they also need some inorganic ions to facilitate processes within them to function well. These are usually in trace amounts.
Both N and P are carbon sources, so in theory they can't be termed inorganic, but somehow when applied to bio-chemistry, they imply to indirect ways of some living things to obtaining carbon.
So inorganic N here, in reefing at least, refers to gaseous N or dissolved Nitrogen. When we say cyanobacteria is N-fixing, it implies that cyano is able to draw, dissolve inorganic N in water to meet their N demands. i.e. you may have very cleans waters with little NO3/NO2 whatsoever, but still hv cyano.
Inorganic-P, refers to water-soluble P or orthophosphate or simply PO4... some ppl wont' accept the definition of orthophosphate as inorganic, because they are the very basis of all building blocks of life, and every life needs it, so much so that there's very little of it... and so perhaps thats where the 'inorganic' part comes in... trace amounts.
HTH, I may be wrong, its my understanding oif it anyways.