Liam Fox got one thing right. As the media hoo-ha grew, he, or someone on his behalf, anonymously warned the Prime Minister that to sack him would be "weak". This was the opposite of the assumption during the Blair period. Alastair Campbell pointed out last week that he never formulated a law that said "anyone who was on the front pages for 10 days in a row was a goner". But he and Blair were obsessed with showing "strength" and "grip", and they would never have allowed the Fox story to run for so long.

"What I think I said," said Campbell, "was if the same story ran for 10 to 12 days, you knew you had a real crisis management issue not a frenzy." Same difference. It is interesting that David Cameron refuses to be the heir to Blair in the way he handles such things. His spokespeople told journalists he didn't want to lose Fox, as well as the standard "the Prime Minister has full confidence", which is usually mocked by journalists as meaning that the subject is doomed. It means no such thing: we just remember the times when it is followed by a resignation and forget all the times when the minister concerned, such as William Hague or Kenneth Clarke, is still there months later.

Cameron has lost only two cabinet ministers, and one press secretary, in his first 17 months. In all three cases, he showed loyalty and restraint. In the case of David Laws, Cameron had decided that he could stay. The first the Prime Minister knew that Laws was going to resign, I was told at the time, was when Downing Street officials tried to call Laws to tell him that he was safe but could get through only to voicemail.

Contrast Cameron's demeanour with the panicky shortness of breath that would have infected No 10 in Blair's time from the moment – nearly two weeks ago – that the Fox story started to dominate the front pages. So calm was Cameron, that I made the mistake of assuming that Fox would survive, just as Hague, Clarke and Chris Huhne had. (In Huhne's case we need to add "so far", but I recall predictions that he would be out by the end of July.) The Guardian reported in August that Fox's friend had been handing out business cards describing himself as an adviser to the Secretary of State, a story which was picked up by The Mail on Sunday. There did not seem to be much that was new about it two weeks ago, apart from tasteless innuendo about Fox's sexuality, in the Mail titles. It should, however, have been apparent that it was a "crisis not a frenzy" on Monday last week, when the Ministry of Defence published a detailed list of meetings between Fox and his friend, Adam Werritty. Plainly, Fox's intention was to get everything into the public domain, even if it contradicted what he had told the Commons the previous month, namely that Werritty had "not travelled with me on any official overseas visits". He was trying, too late and too little, to follow the Alastair Campbell media handling guide, which is to make a full and voluntary disclosure rather than have the embarrassing bits dragged out day by day.

But the ministry's disclosure was incomplete, because there were further details about where the money came from to pay for Werritty's "not travelling with me on any official overseas visits" that were left to the newspapers to report last week. At which point Fox was left to draw his own conclusions. The Prime Minister, with respect to due process and a fair hearing and the fact that anyone can print themselves a business card these days, had given him time by asking the mandarins to have a look into it. One of the sub-mandarins had sat Fox down and asked him some of those questions that barristers do on the television where it looks simple but it's really clever, such as "Where did the money come from?".

So there is a tendency to assume that ministerial resignations are forced merely by the level of hoo-ha that the press can crank up – which is why Alastair Campbell feels misquoted. His "10- or 12-day rule" implies that, if the press can keep up a furore long enough, it will reach the critical intensity that will force the Prime Minister to pull the plug. But it has usually not worked like that, either under the panicky Blair or the unflappable Cameron.

In almost all cases, the minister has broken some rule or other, and the only judgement is whether the breach is serious enough to warrant resignation. At the margin, this can be influenced by the sheer volume and persistence of media hounding. Just as ministerial chutzpah can help to fend off the chop for less serious offences. Peter Mandelson (the second time) and Stephen Byers resigned for minor infractions, but they had had enough (in Mandelson's case, as it turned out, rather briefly). Fox put on a good show of brass neck in the Commons on Monday, persuading many on his side that he could see it through. But usually ministers go for good reasons. David Laws, whatever the extenuations, had wrongly claimed £40,000 of public money. Fox had failed to disclose donations to an associate who seemed to be doing deals or, as we now call it, who seemed to be engaged in "transactional behaviour".

He was not brought down by a media neocon hunt, as some of his allies seem to believe. He decided to go because he realised that the civil service inquiry into his affairs was going to reveal a clear breach of the ministerial rules. The big difference between Cameron and Blair, though, was that Cameron showed the strength to wait for the facts and to allow Fox to reach his own conclusions.

The other big difference was in the handling of the reshuffle. It was notable, although it is a story for another day, that all the promotees are George Osborne's allies: Philip Hammond, Justine Greening, Chloe Smith, Greg Hands and Sajid Javid (who becomes the Chancellor's parliamentary private secretary). Cameron is strong enough to let his Chancellor run his political operation. The TB-GBs are finally dead.

twitter.com/JohnRentoul