The Shagya is not a strain, it is a seperate breed, an Arabian crossed with local mares, started in Hungary, to produce a carriage and military horse with the pretty of the Arab, but slightly bigger and heavier.
As for Polish Arab and Arab....any Arabian with documents from a WAHO (world Arabian horse association) acknowledged Arabian registry is a purebred Arabian.
Poland, Russia for example, did their selection through racing, Germany tested all the stallions under saddle, jumping, dressage, cross country, against warmbloods, before allowing them to breed. Spain bred for Spanish looks, the Andalusian neck and movement, and so on. You used to be able by looking at an Arabian where he came from. Nowadays that is rather rare.
Unfortunately people have forgotten what an Arabian used to be, hardy, sound, people friendly intelligent, a horse who can do anything with grace, style and beauty. Now it's all about pretty for the halter show.