I'm just going through the process of getting a mooring at the moment and I agree, Maritime's website is confusing. Hopefully this'll help sort it out for you.
Basically, there are 4 kinds of moorings:
1. public - few and far between. They are marked on the map, often pink in colour (but the map should tell you what colour they are) and usually only exist in places like national parks to allow people to stay overnight. In areas like the Spit Bridge, there are a few to allow people to wait for the bridge to open. In all cases, you can only stay for a short time (national parks ones are maximum 24 hours)
2. commercial - belongs to a business and is used as a spot to plonk a boat that they are working on. More often, it belongs to a marina and is rented out to the likes of you and me at high prices. Some belong to yacht clubs for the use of their members (only). I think these are orange in colour
3. emergency - blue in colour. Stay off them. The water police and NSW Maritime use them to plonk boats on in an emergency and they'd be sure to get snippy if they found your boat already there
4. private - exactly what it says - privately owned by someone so they can park their boat on it. These are the bulk of the moorings that you'll see and they are yellow. The owner has to buy the mooring equipment and pays someone to have it installed. Usually, it'll be more than adequate to cater for
their boat but there's no guarantee it'll cope with
your boat. The owner has to pay NSW Maritime an annual license fee just for the privilege of taking up a space on the water. Also, while the owner should pay to have the mooring serviced regularly, there's no guarantee that they actually do that so the mooring could be dodgy. So be careful if you decide to pick someone else's mooring.
The process of picking a spot is easy enough. NSW Maritime has a map on their website showing all the mooring areas. They also tell you how many people (if any) are waiting to get a mooring in each area. When you apply, you pick an area. If there isn't any space available within the area you chose then you go on a waiting list until a space does become available. Once a space is available NSW Maritime will then tell you the exact spot within that area where your mooring has to go based on the size and type of boat you have. As with everything to do with the government, there are fees for applying, fees for going on the waiting list, fees for keeping a mooring, fees for scratching your ar$e... you name it.
Lastly, I've found that it's a good idea to ring and talk to the mooring people before putting in an application. In my case, I picked a spot that their website said had no waiting. When I put in my application, however, it turned out that the last spot had just been allocated. So now I'm #1 on the waiting list

If I'd known that was going to happen I might have picked another spot just up the river a little further.