Or Rabinowitz' 2014 book Bargaining on Nuclear Tests argues strongly that the Vela Incident was an Israeli test that was coordinated with or approved by the South Africans, and also asserts that the device tested could not have been a South African device, as they did not yet have one that could have been tested. The book cites NSC and CIA documentation indicating that the US government believed Vela had been a nuclear test and was likely to have been an Israeli test; additionally that the American Naval Research Laboratory produced a report in 1980 for the White House that argued that it had been a small nuclear weapons test. The book also points out that Carter's personal diary published in 2010 has an entry in September 1979 that indicates that he believed it to be a nuclear test.

Rabinowitz cites historian Richard Rhodes and journalist Seymour Hersh for further confirmation of this claim, and notes that a South African government minister in 1997 said that the Vela Incident had been an Israeli nuclear test before walking back that statement.

She argues that the reason why it was officially held to be non-nuclear, with that conclusion remaining the official claim of the US government today, is that a public finding that it had been a nuclear test would have forced Washington to confront Israel at a moment when that would have been a devastating blow to Carter's attempts to carry over the accomplishments of the Camp David accords into the incoming Reagan Administration, and the Reagan Administration similiarly had no interest in forcing a confrontation with Israel.

Rabinowitz argues that Israel took the risk of forcing a confrontation with Washington over a test because the device tested was a low-yield design for a tactical nuclear weapon that might have battlefield use, which had become central to the imagined use of nuclear armaments in Israeli military planning, a type of device that she argues absolutely requires testing compared to basic first-generation nuclear bombs.

Or Rabinowitz, Bargaining on Nuclear Tests: Washington and its Cold War Deals. Oxford, 2014.

The incident:

The US government formed a commission with experts like Luis Walter Alvarez. The commission doubted the validity of this measurement. During US Air Force flights over the alleged explosion area, no corresponding radioactive traces could be found. One explanation proposed by Alvarez was the impact of a micrometeorite on the satellite. Critics, however, see the results of this commission as biased, as the then President Jimmy Carter was against the proliferation of nuclear weapons. A nuclear weapons test, especially with Israeli participation, could thus have become a political problem.

If the Vela incident was indeed a nuclear weapons test, South Africa, whose government had embarked on an ambitious nuclear weapons programme in the 1970s, or Israel are considered the most likely culprits. Apparently, South African navy ships were also in the vicinity of the explosion site at the time in question. According to Sasha Polakov-Suransky, Israel offered to buy nuclear weapons in 1975 from South Africa, with which it also cooperated on other armament projects; however, the then Israeli Defence Minister and later President Shimon Peres denied this.

History as context:

The Carnation Revolution in Portugal on 25 April 1974 was an external impetus for changing power relations in southern Africa. The independence of the former Portuguese colonies Mozambique and Angola in 1975 suddenly confronted South Africa with neighbouring states that openly opposed the apartheid regime, at least by supporting the ANC. Until then, the two Portuguese colonial territories had been foreign policy buffer zones for South Africa's neighbourhood policy, with which it had maintained close military relations.

After the outbreak of the civil war in Angola, the South African army, with the approval of the USA, marched into southern Angola on 23 October 1975, where it also encountered Cuban troops. One of the objectives was to simultaneously fight SWAPO, which had taken up armed struggle against South Africa in 1966 and was operating from Angola. After the withdrawal from Angola, South Africa continued the war against the neighbouring country from occupied South West Africa.

South Africa responded to the foreign policy threats with its own nuclear weapons programme. Since January 1978, the enrichment plant in Pelindaba supplied weapons-grade uranium, from which South Africa constructed six operational nuclear weapons. Defence Minister Pieter Willem Botha had supported a nuclear weapons programme and the military's preparations for a nuclear test during his tenure. German research institutions were also involved in South Africa's nuclear technology development. South Africa used a process that was a modified version of an isotope separation process developed in Germany. Scientific publications on this were also available to the South Africans. Supplies came from US, French, German and Swiss companies. Cooperation with German partners was of considerable importance. Botha became prime minister in 1978.

Conclusion (my own opinion):

Is there a possibility for a Secret Test? Yes. Is it historically logical to conduct a test at this time and place? Questionable. There was a nuclear weapons programme at the time and the technology, but it is questionable whether the two parties would dare to do so. I don't want to start speculating in this Reddit and leave it at that, that there was a possibility, but it still has to be looked at critically. Everyone should form their own opinion.

I hope this has been helpful to you. If you have any comments, questions or clarifications, please post them in the comments.