It's not. Here's what I see if I go to book a flight on Icelandair's website:
The premium cabin is sold as "Saga Premium," as also shown on their services page. Their fleet pages depict "Saga Premium" and "Economy" classes, not business. Which strikes me as fair. It's clearly a premium service as compared to economy, even if there aren't all the benefits ...
The Qatar Privilege Club earning chart indicates the following:
Flexi: J, C
Flexi: Y, B, H
Value: K, M, L, V
Saver: S, N, Q
Promo: T, O, W
Group or Tour: G
I spoke with BA over twitter.
The code would relate to the rules and restrictions on the ticket. If
booking via our website, these would be advised at the time of
booking. Nearly all flights would have underlying fare types such as
this, and as you point out, 'A' fares would usually be restricted
First class fares. These would normally have fees ...
If the itinerary is booked on one ticket (its a code-share), then the Most Significant Carrier rule (from IATA) applies.
Air Canada has a pdf on their website that details rules and conditions of carriage, here are the relevant bits:
In the case of code-share, the baggage rules of the first marketing
carrier (carrier whose code appears on the flight ...
The important external thing (for naming a booking class "Business class") is whether airports recognize it as business class (and get the appropriate fees for those passengers) and hence honor their business class services to those ticket holders. Then the airline can call a ticket business-class. What happens inside the airplane is really the airline's ...
As fare codes, after the first letter, are airline specific and BA does not share the full list with the general public, you may never find the full explanation. You worked out much already by working out you need to stay a Saturday night.
It is a long and complicated code but those are not unlikely, even though they are rare.