Mr Cameron said he decided that his remarks, delivered in 2010, needed to be “absolutely direct and clear”.
Another man shot by paratroopers on January 30 1972 died four months later. While many consider him the 14th victim of Bloody Sunday, his death was formally attributed to an inoperable brain tumour.
After Lord Saville produced a report in 2010 which stated that none of the casualties were posing a threat or doing anything that would justify their shooting, Mr Cameron apologised in the House of Commons, saying that the killings were “unjustified and unjustifiable”.
Speaking about the events of that day, he told the BBC Radio Ulster Talkback programme that the Saville report was one the “most shocking things” he had ever read.
He said: “What I was feeling was what a huge responsibility it was to try and get this right, because the families affected, people in Northern Ireland had been waiting for this for so long.
“Not just waiting for the report, but waiting for a sense of justice and recognition so I knew it was a huge responsibility.
“The way I approached it, I was in Afghanistan visiting the troops and I came back and was told the report is now ready.
“I remember my office being crowded with people all wanting to talk about how we should respond and I just wanted to chuck them all out and read the whole of the summary on my own with no interference so I can really see what Saville is saying.
“And it was one of the most shocking things I have ever read and I knew before I read it what a responsibility it was, but I knew in reading it something very special, very clear, very frank needed to be said.”
Mr Cameron continued: “I knew in this case there would be thousands of people listening in Derry wanting to know what Saville had said.
“Ultimately I had come to the conclusion that it needed to be a very clear and frank apology and explanation, and don’t try and qualify it in any way.
“The one thing I tried to do was to explain why I felt so shocked by what I had read.
“The shocking nature of what Saville uncovered, I just knew it had to be very clear, very frank and no trying to have it both ways.”
Explaining his use of the phrase “unjustified and unjustifiable”, he said: “It was what I felt when reading the report.
“Sometimes with these sort of events, politicians – and I can be guilty of this – qualify the apology or shade around the edges.
“On this occasion it needed to be absolutely direct and clear.”
Mr Cameron said he was “very moved” when he later saw the scenes in Derry of people welcoming his speech.
Lord Saville told the programme that the question of whether there should be prosecutions for what happened at Bloody Sunday in 1972 remains “difficult”.
He said: “I can understand completely the feeling of the families of those who died that they want to see justice done, that is a perfectly reasonable position to take up.
“On the other hand, decades and decades have gone by and it can be said that it is really rather unfair to soldiers to prosecute them after such a long time has gone by.”
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