I think coach is more a long distance thing and bus is a more round about town sort of thing. I suspect when people in the USA say coach they are thinking of Greyhound which is a bit like National Express but slightly less good. I have never thought of coaches or buses as "economy". In Edinburgh, buses are more a practicality (or they were until they closed the city to tram works).

Back on topic, though, I noticed that First Scotrail had changed the name of their business class sections back to first class. Common sense prevails. Do any other rail operators call first class business class?