Origin of the titles "Doctor" and "Master" from Doctor Who

Question

First off, I do not remember the episode but I do recall that it was either during the Eccleston or early Tennant eras.
The Doctor was talking to someone (Rose?) about his days at the Academy where the other children called him The Doctor because he would help those that were hurt. Meanwhile, the title of The Master was given to one of the children because he would always tell others what to do.
Were there other titles given to other children at the Academy like the Doctor and Master?

Answer 1

Most Time Lords seen during both the classic episodes as well as the new series seem to have adapted to using these titles with a few rare exceptions, like Rassillion. Some are even clear titles being assigned, like President. Although - as noted in the comments - Classic Who has a significant number of Time Lords addressed by name, too. As such this impression might actually be wrong.

While I can't remember details about the Master, it's revealed or at least heavily implied/suggested by River Song (Series 6, episode 7: A Good Man Goes to War) that at least the Doctor's name is in itself a self-fulfilling prophecy/loop:

This was exactly you. All this. All of it! You make them so afraid. When you began all those years ago, sailing off to see the universe, did you ever think you'd become this? The man who can turn an army around at the mention of his name. "Doctor": the word for "healer" and "wise man", throughout the universe. We get that word from you, y'know. But if you carry on the way you are, what might that word come to mean? To the people of the Gamma Forests, the word "doctor" means "mighty warrior". How far you've come. And now they've taken a child, the child of your best friends, and they're going to turn her into a weapon, just to bring you down. And all this, my love, in fear of you.

• The Doctor got this name for acting like a doctor or healer, helping others with their issues.
• Doctors throughout the galaxy adapted that title because of how the Doctor acted.
• It's also noted by hear that at some point people started to associate the name with being a great warrior, too, although that didn't last.

Answer 2

Since @Mario has addressed the question in-universe, I thought I'd discuss the real-world origins. As Wikipedia notes with a now invalid link to a 1970 internal memo,

The Master's title was deliberately chosen by producer Barry Letts and script editor Terrance Dicks because, like the Doctor, it was a title conferred by an academic degree. The brief given in a sketch of the three "new characters" for 1971 (the other two being Jo Grant and Mike Yates) suggested he was conceived to be of "equal, perhaps even superior rank, to the Doctor".

Wikipedia also summarises The Handbook: The First Doctor – The William Hartnell Years 1963–1966:

The character of the Doctor was created by BBC Head of Drama, Sydney Newman. The first format document for the programme that was to become Doctor Who – then provisionally titled The Troubleshooters – was written in March 1963 by C. E. Webber, a staff writer who had been brought in to help develop the project. Webber's document contained a main character described as "The maturer man, 35–40, with some 'character twist.' " Newman was not keen on this idea and – along with several other changes to Webber's initial format – created an alternative lead character named Dr Who, a crotchety older man piloting a stolen time machine, on the run from his own far future world. No written record of Newman's conveyance of these ideas – believed to have taken place in April 1963 – exists, and the character of Dr Who first begins appearing in existing documentation from May of that year.

This suggests "Doctor" was chosen to imply some kind of well-informed, aged seniority, reflected in an academic title.