Since we can debate, I'd like to offer a globist standpoint.
I was surprised that nobody said it. The zetetic method is the scientific method. There's no difference between them. That's because modern science is based on zetetic.
It started with Descartes, genius mathematician and philosopher, father of rationalism. His main principles were logic and reason. He seeks the truth of knowledge through strict analysis. He rejects all faith and preconceived notions in science, although he is a fervent believer.
The need for a rigorous method to acquire evidence, the falsifiability of hypotheses, the reproductibility of experiments, the skepticism against everything that isn't fully demonstrated: all of this is from Descartes 'Zététique'. Which greatly inspired and allowed Newton to develop mechanics.
His most famous result is probably 'Cogito ergo sum': I think, therefore I am. This single thought experiment kick-started the age of knowledge we live in, by putting doubt at the base of all science.
Recognizing that everything must be doubted, including his own intelligence, his knowledge and his senses, which are mere representations of the world in his brain, he searches for one undeniable truth of fact. This truth should serve as a foundation for a new kind of knowledge, acquired exclusively through reason, after this thought experiment.
And he finds that no matter if his intelligence and sensations are false, he can always feel his own mind talking inside himself, which implies that 'himself' must somehow exist: he thinks, so he exists. From the truth of his own existence, he then goes on to recover knowledge, by analysing his previous ideas and keeping only those rationally demonstrable from the Cogito.
As such, zetetic is an important course taught in many reputable and scientific universities worldwide. It's a tool for mental hygiene. In fact, it should be in the toolbox of every self-respecting scientist to regularly get back on track. What are my assumptions? How strict is my method? Are my results sufficiently demonstrated? Am I not influencing the experience with my own bias?
Flat Earthers don't use the zetetic method but a parody of it. They don't use doubt to gain knowledge. If that was the case, they would perfectly accept scientific findings, precisely because scientists are the most skeptical persons in the world.
Instead, they use doubt to remain forever in doubt. They refuse to leave the Cogito and build further knowledge. Which results in the perpetual demands for more evidence, meanwhile rejecting all evidence presented.
Pseudoscience groups, along with paranormal, spiritual, and all kinds of alternative groups, have appropriated zetetic to mean: 'we doubt science'. Forgetting that science is the method of doubt.
What they mean by zetetic depends on each group. They don't always replace rationalism with empiricism, but this seems to work for Rowbotham's business as a reason to distrust technology.
If proof was needed that Flat Earthers don't practice the zetetic method: they're seldom skeptical of each other. Any evidence against science triggers a confirmation bias, whether the actual belief in this evidence is shared or not. Meanwhile scientists spend their lives attempting to destroy each other's theories.
Whatever definition these modern groups give to zetetic in pop culture has little influence on the academic world, which still relies on the zetetic method.
I know some will disagree and hopefully I don't appear aggressive. It's a shame to read that Occam's Razor isn't a part of zetetic. In the real world, it is.