The definition of 'satsuki' I like most is any azalea that is registered with the Japanese satsuki society (and maybe first generation offspring as well). But the question was how to tell the difference. I am not really sure how I am able to tell the difference. It is almost like asking someone about how they can tell the difference between the face of their younger and older brother. If you see them, you just know, because of experience. You may tell someone 'my older brother has a larger nose', and that may be true. But if you recognize them from their faces, you don't consciously look and measure the size of their nose.
Very generally, there are two types of satsuki. One that resembles R.indicum. And one that resembles R.tamurae (or aka R.eriocarpum,. It is easier to recognize a R.indicum-type satsuki. And recognizing either of those is easiest by linking the azalea hybrid back to either R.indicum or R.tamurae (or more accurately the archtype maruba satsuki that is a hybrid of R.indicum and R.tamuae).
This is how R.indicum looks:
Source:
https://mstyasou.exblog.jp/19982250/
The narrow leaves you see in this picture are the same as you see in 'Osakazuki', 'Kinsai', 'Korin', 'Kozan'.
The only other azalea that has these narrow leaves is R.nakaharae. I cannot tell the difference between R.nakaharae leaves or R.indicum leaves. It doesn't help that all R.nakaharae hybrids I have seen were R.nakaharae x R.indicum. The way to tell if something is an R.nakaharae hybrid is easy, though. If it is creeping, it has R.nakaharae genes. In that case, you cannot tell if it is also R.indicum/satsuki. If it is mounding, it is R.indicum.
Another way to recognize a satsuki is the elegant round flower shape. This is largely a result of breeding and not of the species. Satsuki were bred as potted plants with individualistic flowers. All other azalea groups are largely landscaping plants, where specific flower shape and petal roundness is irrelevant.
R.tamuarae in the wild looks like this:
Source:
http://samy8940.tea-nifty.com/blog/2012/05/index.html
The leaves are very different, hence the name 'maruba satsuki' aka 'round leaf satsuki'. But, the leaves not very distinct to some kurume. The best way to tell something is a maruba satsuki is actually the flower shape. It is the famous ruffled (namiuchi zaki) shape of Gumpo and others:
So R.tamurae explains the purple colours and the rounder leaves in satsuki. But hybrids of the species R.indicum and R.tamurae actually look nothing like most satsuki.
There are many volcanoes in Japan where R.kaempferi (red flower) and R.kiusianum (purple flower) overlap. And you find natural byhrids of all colours there. and these hybrids resemble kurume cultivar.
R.indicum and R.tamurae actually overlap very little. There is an area in Yakushima island where natural hybrids are found, and they too have a range of colours not seen in the species. But apparently these natural hybrids lack some of the features we see and like in satsuki. You cannot find sporting or the above namiuchi flower shape, so distinct for maruba satsuki.
The hypothesis is that sporting in satuski comes from R.simsii as apparently out of all species forms, R.simsii are most prone to sporting.
Some more reading:
giulioveronese.com
I especially recommend 'The Origin of Satsukis: The Yakushima Connection' by Barry Yinger.
I don't know if this helps. But years of looking carefully at every azalea you see anywhere helps.
There are many hybrids though that elude IDing of species, often because their background is so diverse. If a hybrid is 12.5% R.kiusianum, 12.5% R.simsii, 12.5% yedoense poukhanense, 12.5 R.indicum, 12.5% mucronatum and 12.5% kurume, 12.5% maruba satuski, and 12.5% R.nakaharae, how does it look?