Hi,
It is exactly like you described, one of my customers and a customer of them were victims of that scam 3 years ago, their customer paid around 50 thousand Euros thinking they were paying to my customer, fortunately their customer has insurance and recovered the money.
In our case, there was an additional detail, the scammers asked to make the transfer to a different account regarding to a real invoice they had attached in a PDF file.
So, the suspicious was that e-mails were hacked on one of the sides and the first measure was to change all the passwords.
But you're right, besides the apparently trustworthy information on the emails, the lack of accuracy on the message construction should have been a sign but, it wasn't.
Another unbelievable thing was that Loyds Bank in London didn't check any information to open the account for the scammers, all the details (name, address, VAT number) were correct and identifying my customer.
After that, my customer informed all their customers that any change on their bank account must be confirmed by phone.
I remember these were terrible days, I was in the day before to return to Brazil after a period in Portugal precisely to move this customer from their internal server to a Datacenter and, obviously, all the doubts came up if the Datacenter wasn't the culprit