Emperor - He is not Astartes, Astartes is less than The Emperor as they are made by the Emperor twice removed.
Primarchs are much more than the Astartes too. Astartes are made in the image of the Primarchs.
Its like comparing a copy of a copy of Mona Lisa picture with the original painted by a 10 year old.
In some of the Horus Heresy books it is mentioned that the Primarchs have some other organs not in the "normal" Astartes.
Astartes means Stars, Adeptus means the ones that get hold of something. So the Adeptus Astartes means literally those who conquer the stars aka Space Marines.
By my very literal translation anybody could be an Adeptus Astartes but nobody does it as good or as fast as the Emperors Space Marines.
But in the W40k it means that ordinary humans that survive decades of grueling mental and psycic testing and organ implants get to call themselves Adeptus Astartes.
So if you are unlucky to get selected by a Chapter for induction you have to be mentally and physicaly tough. You have to maintain your levels of excellence and on top of all of that you have to posses some luck as the implanting process doesn't take every time. A lot of recruits actually die in this last stage of becoming an Astartes.