The term renegade Time Lord was an epithet applied by Time Lords against their fellows who went against Time Lord law. At various points, the Doctor was called and thought of himself as a renegade — but he also applied that label to others.
Of the various reasons to call someone a renegade, none was more important than the simple act of physically departing Gallifrey. In this sense, the Seventh Doctor clearly thought of both himself and the Master as "renegades". (PROSE: Doctor Who - The Novel of the Film) There was, in this definition of the word, no connotation of any particular morality, since the Doctor and the Master had generally opposing value systems.
To be labelled a renegade, it was enough to be a Time Lord not residing on Gallifrey, though in some cases a Time Lord could be responsible for some action the rest of the species found abhorrent. For instance, in the cases of two highly respected individuals, Hedin, a member of the High Council, betrayed Gallifrey by trying to aid the long lost Omega, (TV: Arc of Infinity) while the equally respected and renowned Ophiuchus conducted experiments designed to extend the Time Lord regeneration cycle. Accused of crimes such as vivisection, Ophiuchus was labelled a renegade. (COMIC: Ophiuchus)
Indeed, the hard choices involved in actually leaving Gallifrey were, according to the Fifth Doctor, a key distinction between a renegade and a mere "political rebel" still resident on Gallifrey. He felt that Ruath could have been a rebel, but not a renegade. (PROSE: Goth Opera)
However, since Gallifrey had laws against interfering in the natural course of events, and leaving Gallifrey on un-sanctioned voyages typically led to interference, at least some segments of the Time Lord policy thought of renegades as definitionally criminal. For instance, when the Doctor was once accused of being a "dangerous renegade", Flavia instantly snapped to the Doctor's defence and pointed out that he wasn't a criminal. This immediately suggested that the word was commonly used to indicate criminality. (PROSE: The Eight Doctors)
Though the Doctor didn't himself feel that interference itself was illegal, (TV: The War Games) he did freely admit to interfering. Indeed, the Eighth Doctor felt that one definition of a renegade was a Time Lord who regularly picked up people on one world and transported them to another. (PROSE: Legacy of the Daleks) This implied that interference was part and parcel of being a renegade. On another occasion, the Eighth Doctor was pressed as to why he was a renegade. He simply said, "There are times when a little intervention is necessary." (PROSE: Interference - Book Two)
The legality of interference was debatable and the subject of several appearances in court for the Doctor. Out of his three known Time Lord trials, only one ended in conviction for interference. (TV: The War Games) The Sixth Doctor's trial ended in dismissal, (TV: The Ultimate Foe) while the Fifth Doctor escaped for unknown legal reasons, but not before being accused of playing too much cricket and essentially not interfering enough. (COMIC: The Stockbridge Horror) Even the Doctor's one conviction was hardly carried out with conviction. The Celestial Intervention Agency were only too happy to use the convicted Second Doctor to carry out their plans. (PROSE: World Game) Even the Third Doctor's so-called "exile on Earth" involved semi-regular contact with Gallifrey. (TV: Terror of the Autons, Colony in Space, The Curse of Peladon, The Mutants, The Three Doctors)
Thus, violation of the non-interference policy, while a frequent characteristic of the renegade, was simply not prosecuted consistently enough to be the determinative reason someone was labelled a renegade.
On still other occasions, other kinds of criminality engendered the use of the term. For instance, the First Doctor condemned Borusa as a renegade because his former teacher had broken any number of laws in his foolhardy search for immortality. (TV: The Five Doctors)
Renegade Time Lords could choose to undergo the Elective Semantectomy to remove their name from history and replace it with an epithet (like "the Hussar"). This was primarily performed as a way to protect the reputation of one's house and bloodline. (PROSE: Weapons Grade Snake Oil)
During the War in Heaven, the ruling Houses promised pardons, political power, and new regeneration cycles to renegades in an attempt to bring them back to the Homeworld. (PROSE: The Book of the War) Some were reintegrated into society; Holsred's biology tutor and Allopta's combat teacher were barely-reformed renegades. (PROSE: The Taking of Planet 5) Others were betrayed, overpowered, lobotomised by military psychosurgeons and fitted with neural links to babels for the Lethean Campaign. (PROSE: The Book of the War)
In the case of two renegades, the other Time Lords were known to intervene to extend their lives. In the case of the Master, as well as resurrecting him to fight in the Last Great Time War, (TV: The Sound of Drums) the Time Lords granted the Master a new regeneration cycle. (TV: Utopia, The Doctor Falls) Though this particular intervention was to their advantage, the Time Lords later cured the "condition" the Master had developed from his botched resurrection and exiled the Master from Gallifrey rather than killing him when it was of no advantage to them. (TV: The End of Time, The Doctor Falls) In the case of the Doctor, the Time Lords intervened to save his life at Clara Oswald's request by granting the Doctor a new regeneration cycle when he was at the end of his life, altering the future and breaking their own non-interference policy. (TV: The Time of the Doctor)
Three renegade Time Lords, the Doctor, the Master and the Monk, were known to have survived the apparent destruction of Gallifrey at the end of the Time War, (TV: Utopia, AUDIO: Divorced, Beheaded, Regenerated) remaining in N-Space while Gallifrey was hidden in a pocket universe. (TV: The Day of the Doctor)