Do the Doctors in the Doctor Who series identified themselves well?

Doctor Who refers to how characters identify the David Tenant character and past actors who have played the time lord.

Doctor is the title that the time lord uses to identify himself, and this started way back in the first Doctor Who film back in 1965 called Dr. Who and the Daleks. In the beginning, the Doctor reference was regarded as a Ph.D. title since the time traveler was an inventor who invented a time machine, and went on an adventure with his family. In the film his last name was actually "Who".

The film versions are slightly different from the original 1963 television shows that introduced the time lord. There he is referred to as "The Doctor".

It's only later that the BBC combined the two to create the name "Doctor Who" as the main character, rather than a show title.

The only other time lord to be featured in the series is the The Master. So they are not referred to as "Doctors". Only the main character is Doctor Who.

To refer to multiple time lords you would call them Gallifreyans, since they come from Gallifrey.

The Doctor does have a real name, but it's never been revealed in the show. That's part of the mystery of the character.