How did Jean Parisot de Valette, 49th Grand Master of the Order of Malta, die?

The perceived discrepancies in question are not necessarily all too contradictory. Most accounts repeated elsewhere of this are very short and just leave out specifics. Further, earlier understandings of the medical reason side were probably hampered by basing a diagnosis on outdated medical knowledge.

If he went hunting and got a sun stroke, he didn't have to die immediately on the hunting grounds. We now know it can take quite long for the effects to result in death:

It was long believed that heat strokes lead only rarely to permanent deficits and that convalescence is almost complete. However, following the 1995 Chicago heat wave, researchers from the University of Chicago Medical Center studied all 58 patients with heat stroke severe enough to require intensive care at 12 area hospitals between July 12 and 20, 1995, ranging in age from 25 to 95 years. Nearly half of these patients died within a year – 21 percent before and 28 percent after release from the hospital.
WP: Heat stroke

One widely cited author writes on de Valette

In July 1568, three years after the siege, La Valette suffered a stroke from which he never recovered. His last few years had not been particularly happy. Apart from the pleasure of seeing his new city rising in white limestone blocks under the brilliant southern sun, his time had been taken up with resolving innumerable disputes among the young Knights who, when not absent on their caravans, found time heavy on their hands in the soft and indolent atmosphere of Malta.
_{Ernle Bradford: "The Shield and the Sword. The Knights of Malta", 1972.}

While the German Wikipedia simply asserts:

Jean Parisot de la Valette […] died on 21 August 1568 during the prayer in the Fort St. Angelo in Birgu. Today it is believed that he died of heat stroke after a day of hunting.

But the source for that information is neither given directly, nor found in the references cited on that page.

The one source German Wikipedia gives just notes

J.P.V. died on 21 August 1568, a good three years after the siege, from the consequences of a stroke. _{Tade Matthias Spranger: "Jean de la Valette". In: Biographisch-Bibliographisches Kirchenlexikon (BBKL). Vol 17, Bautz: Herzberg 2000, Sp. 1448–1452.}

A recent and very detailed account gives a convincingly complete picture:

What dealt him the last cruel blow was the murder, on July 31, 1568, of a young woman, very likely his illegitimate daughter. He had held Isabella Guasconi in baptism “because of a personal obligation” – (the statutes forbade knights acknowledging illegitimate offspring, and they circumvented this ban by standing as godfathers). This bellissima giovinetta married a Florentine, Stefano Bonacorsi. Claiming she was cheating on him, he murdered her, absconding immediately with all her jewellery and precious objects.

De Valette lost his mind in grief. He convened the Council in a matter with which the Council had nothing to do – neither the criminal nor the victim were members of the Order. This episode perturbed him profoundly “sentí un cordoglio grandissimo”. He never recovered. Within a couple of days he was seriously ill and three weeks later he died.

After Isabella’s murder, de Valette needed desperately some distraction to push back his overwhelming sadness. He went to St Paul’s Bay, hunting for partridges with falcons. “The sun shone so ardente e piccante, that neither hats nor umbrellas could prevent it going through his brain and addling his mind. Next morning, wanting to hear mass in the palace chapel, he was suddenly attacked by a fiero e terribil accidente, that hurled him to the floor. Having been straightened and carried to a bed, the faint receded, but a double terzana, took over and for four or five days so increased in fury, that the doctors started fearing for his life. Then the fever changed to a simple terzana and mitigated its rigour, so that it gave hope of a substantial recovery and health”.

In this condition, news reached him that the Turkish fleet had attacked Calabria and headed for Malta. The Grand Master summoned the elders to his bed chamber. He explained his state of health, delegated all his powers to the Council, and asked to be taken to the new city (Valletta) to be close to where things were happening.

On August 5, he seemed to be recovering sufficiently and asked permission to dispose of some slaves. Five days later he appointed Claude de Glandaves as his lieutenant and plenipotentiary, realizing that his illness would not be short. He granted a general amnesty to all knights who had been convicted and deprived of the habit.

Though hopes for his recovery increased, on August 16 he was assailed by “a continuous and slow fever, which, working malignantly on the inside, was hardly perceptible on the outside”. He now understood that his malady was mortal and sent for his confessor, although he had just confessed when he has struck. He got up from bed and knelt on a prie-dieu, “covered with velvet cushions” Bosio hastens to add. He received the Eucharist, then returned to bed “full of spiritual comfort and restored with corporal food”, giving himself up to his doctors, as he entertained no further hopes for recovery.

He then made his last will, leaving the ‘disposable portion’ to his new church of Our Lady of Victory in Valletta, which he had ordered to be built, and to his family and persons he loved, the rest, to the Order. He called the Councillors, embraced them all and asked forgiveness for any failing. He appealed to his nephew Cornisson “to accept God’s will and not to be saddened by his death”.

A multitude of sinister omens gave warning of his imminent death. A mysterious and deafening noise was heard in the sky “horrid … like a great concert of arquebus guns”. A large school of dolphins ran aground in Marsaxlokk. Then all his pets died together: his red parrot of which he was extremely fond, so red it looked like a ruby; his griffin, a personal gift from the King of France, and his lioness, so tame she often slept in his bedroom. The griffin and the lion stood as his heraldic symbols on his coat of arms. With all these portentous signs, the end was inevitable.

De Valette asked for extreme unction, and for a crucifix which contained a fragment of the vera crux. He kissed it repeatedly, all the time imploring the mercy of God, in between sospiri e sussulti. Without ever losing consciousness, he turned his eyes to heaven and said “Elas, mon Dieu, envuoye moy un de tes bons Anges, qui m’assiste en ceste extremite”.

Uttering Giesu Maria he died on August 21, three hours after sunrise, exactly on the anniversary of his election as Grand Master.

After the ritual cutting open and embalming, the body was dressed in the long robe and the manto di punta, and laid in the main hall of the St Angelo palace, on a funeral couch draped in black velvet with gold braid, close to the solid gold sword and dagger sent to him by Philip of Spain after the siege, now in the Louvre. Many torches lit the cataletto. “An infinite multitude of people came to pay their last respects, as he was by those loved like a true father”. Most moved of all appeared some old Rhodian and Maltese women to whom he had daily given alms. They tore out their hair and beat their breasts, raising a lamento compassionevole.

_{Giovanni Bonello: "Grand Masters in the Cinquecento: their Persona & Death", Historical Perspective, Malta Medical Journal Volume 15 Issue 02 November 2003 49. (PDF)}

Jean Parisot de La Valette developed a severe fever the day after he had been out hunting in extremely hot weather and never fully recovered. Valette, who was 73 at the time of his death, was also under considerable stress due to the responsibilities of being the Grand Master (he had been the subject of much recent criticism) and he had suffered a personal tragedy.

Bruce Ware Allen, in The Great Siege of Malta, citing an article in the Malta Medical Journal summarises:

On July 31, 1568, Valette’s (illegitimate) daughter was murdered by her jealous husband. It was a blow from which he never quite recovered. Some days later, while attempting to distract himself by hunting partridge with his falcons, he suffered a stroke....He died on August 21, the anniversary of his election as grand master,…

Roger Crowley, in Empires of the Sea gives a similar account, but mentions heat:

In July 1568, in the heat of a Maltese summer day, Jean de La Valette suffered a severe stroke as he was riding home from a day’s hawking in the woods.

Crawley does not mention sources but Allen's account is based at least in part on the article Grand Masters in the Cinquecento: their Persona & Death by Giovanni Bonello, except that Bonello does not actually say it was a stroke, nor that what ailed him materialised while he was out hunting.

Bonello, who consulted "the manuscript records of the Order of Malta" in the Archives of Malta as well as Giacomo Bosio's Istoria (Vol. III, Rome, 1602) goes into considerable detail (too lengthy to paste here). In short, a combination of criticism of his actions since the Great Siege of Malta, the murder of his (presumed) illegitimate daughter, and the after-effects of an extremely hot day while out hunting, proved to be more than his advancing years could support.

Bonello relates that Valette, to distract himself from the murder,

went to St Paul’s Bay, hunting for partridges with falcons. “The sun shone so ardente e piccante, that neither hats nor umbrellas could prevent it going through his brain and addling his mind. Next morning, wanting to hear mass in the palace chapel, he was suddenly attacked by a fiero e terribil accidente, that hurled him to the floor.

The implication here is sunstroke / heat stroke. Although his condition improved by August 5th,

on August 16 he was assailed by “a continuous and slow fever, which, working malignantly on the inside, was hardly perceptible on the outside”.

Valette realised he was dying, made another confession and his last will. He died on the morning of August 21st.