The screen capture above shows a flat-bed seat even though the flight is marketed as First. You can also look at seat maps. Both delta.com and Google Flights generally get it right. Do you have an example where Google Flights is getting it wrong? It currently seems confused by flat-bed equipment on JFK-SAN/LAS routes and doesn't show the flights at all when searching for FC/Business. If you search for economy fares, it will show the seating type in economy, not FC.