“This TV series should only be considered as a work of fiction.” That was the view of the Versace family before The Assassination of Gianni Versace aired. The FX series, showing in the UK on BBC2, is one of a recent run of shows to dramatise recent history. It focuses on the designer’s murder, on the steps of his Miami mansion in 1997, taking it as the starting point for a nine-part drama that admits to some storytelling licence. Meanwhile, The Looming Tower, which began on Amazon Prime last week and is based on Lawrence Wright’s book, has reopened controversy about 9/11, and in the US, critics are discussing whether Paramount Network’s drama Waco has too much love for cult leader David Koresh.

So, should we view these shows only as works of fiction? The debate boils down to whether the point of fact-based dramas is to reveal what really happened, or if it is to tell a wider dramatic truth. That’s tricky, because the answer is almost always: both. This means that, callous or unethical as it might seem, it’s sometimes OK for such programmes to ignore complaints about events being reshaped for artistic purposes.

A stark and difficult example was The Secret, ITV’s grim 2016 dramatisation of the murder of Trevor Buchanan and Lesley Howell by their adulterous spouses. The complaints from Lesley Howell’s daughter that her mother had not been portrayed correctly deserved, of course, to be heard – but they didn’t deeply affect the validity of the series. That was all about telling the story of Colin Howell (James Nesbitt), a predatory egotist for whom religion offered justification for pursuing his sexual urges at all costs. Dramatically speaking, it didn’t matter if his victims, who didn’t feature heavily, were not presented with precise accuracy.

In contrast, it did matter that The Curse of Steptoe, a BBC4 biodrama aired in 2008 but then withdrawn from circulation in 2010 after a damning BBC Trust investigation, drew flak from relatives and colleagues of Harry H Corbett and Wilfrid Brambell: among their assertions was that the central point about Brambell and Corbett hating each other and subsequently feeling cursed by Steptoe and Son was false.

The Assassination of Gianni Versace’s showrunner, Ryan Murphy, is acutely aware of the dangers. His series from last year, Feud: Bette and Joan, was a fabulously gossipy take on a bygone Hollywood era, only slightly marred by the fact that one of the supporting characters, Olivia de Havilland, is still alive and is now suing. That’s annoying for Murphy, but for us, Feud’s smart take on the lonely, bitter showbiz world isn’t diminished. Murphy is brilliant at finding the story beneath the story; the moral that renders exact truth irrelevant.

The Versace story is the second series under Murphy’s American Crime Story banner. The dramatic truth of the first, the Emmy-winning The People vs OJ Simpson, was that the Simpson trial illustrates how modern America’s ravenous 24-hour media and ingrained racism interlocked in the 1990s and are still a twin menace today. The Assassination of Gianni Versace doesn’t make as profound a point about society, although a theme develops about how the celebrated designer had a different experience of being gay in the homophobic 20th century US to lonely dropout Andrew Cunanan. It is primarily a chillingly authentic portrait of Cunanan, a fantasist who became a serial killer.

In that sense, the show is roughly in the same category as The Secret. The Versaces’ complaints also feel akin to those made by the Matthews family about The Moorside, BBC1’s 2017 dramatisation of the fake “disappearance” of Yorkshire nine-year-old Shannon Matthews. Her relatives didn’t want the story to be examined afresh and thought they ought to have been consulted, but for viewers there wasn’t a compelling reason to heed them. Similarly, to the extent that The Assassination of Gianni Versace is even about the Versaces, it’s about analysing their effort to maintain and control their public image. Seeking their approval would undermine that.

In any case, New York Magazine’s online culture site Vulture has been fact-checking each episode, with help from a reporter who covered the Cunanan story at the time. The embellishments and elisions they’ve turned up have been no more than an intelligent viewer would expect from a fact-based drama. The genre creates its own unique suspension of disbelief, where what we see is both real, in the sense of being plausible or instructive, and not, because we know that some of it has been invented.

Amazon/Hulu’s political expose The Looming Tower, starring Jeff Daniels as John O’Neill, the FBI agent who tried to bring down Osama bin Laden before he committed a major atrocity, illustrates the point in a different way. Daniels plays a whisky-downing, ursine veteran who is juggling two mistresses and rubs the stuffed shirts in the CIA up the wrong way. He’s too much of a boilerplate flawed/ambiguous hero, in other words. Even if O’Neill really was like that, it doesn’t feel true. The plotting of Bin Laden and his associates, and the field work by US law enforcers trying to break these terror networks, is also too laden with familiar spy-thriller devices to be convincing. Lots of what we see might well have happened, but since The Looming Tower all has the air of contrived fiction, it doesn’t matter, because we don’t feel as if we’re witnessing reality. That’s a very delicate balance, but it’s the one Ryan Murphy has mastered.

Gianni Versace and The Looming Tower: does truth matter in ‘true-life’ dramas?