Yes, both Bulgaria and Cyprus are without any doubt part of the west. Both are even part of the European Union. Claiming that they are not part of the west is laughable. Bulgaria is adjacent to Greece which is also part of the west. Culturally large parts of the Balkans are also part of the west.
That's news to my family in Cyprus who very much think of themselves as living in the middle east.
Since you also think that Poland is not the west this does not surprise me that your family thinks so.
This thread speaks for itself. So I'll leave it at that.
Yes, it really does. It's not like the Warsaw packt was named after a town in Poland or anything.