I wonder why I get as another translation exactly the same one I wrote
The only difference I see is that there is one extra space in their answer.
could we say "Kyrkan ligger öst om staden"?
No not really it would only sound wrong and unnatural you should say "Kyrkan ligger öster om staden"
I don't think this sentence is correct
It should say :
The church is in the east of the city
The church is east of town...
Is the word "stad" mean both "town" and "city" in Swedish ?
Yes, that is correct. The word "stad" means both town and city, there is no difference.