When I say iCloud.com I was talking about going to that website with my Macbook and Safari browser. There is where I receive
"iCloud.com Alias cannot be saved at this time" error every time I try.
As alternative to that I tried to do same thing but on my iPhone using iOS mail settings in Settings > iCloud > Mail. Reason why I went that route is because I saw that there is
UI for alias setup in iOS.
Now when I went to that tutorial again I think that I misunderstood that tutorial. I should first go to the iCloud.com website and create alias there and then I will have UI on iOS and macOS to setup alias options the way I want.
Googoling I have stumbled upon this sentence
"Also make sure to check that your region and address information matches the country you’re in. Sometimes those sorts of minutia can lead to issues". I'm currently moving from one country to another and I did switch my region but not my all information is complete yet. I'm also missing payment option since I still didn't open a bank account from new country. Maybe that is a problem.