The first email is correct, you just take your passport and the biometric appointment confirmation. They will check your passport for ID, stamp your biometrics appointment sheet, take your fingerprints and photo and then you leave. Once you have been to the appointment, you place the passport and stamped sheet with your documents and mail them all to Sheffield.
The second email is more generic and doesn't apply to applications made in the US. However, in other countries, you submit your documents in person at your biometrics appointment instead of mailing them.
In the US though, biometrics are done by US immigration on behalf of UKVI, so they don't have anything to do with your application documents - they're just there to take your biometrics and nothing else.