With respect to the Persian king, Encyclopedia of Kitchen History by Mary Ellen Snodgrass, p 427, offers (sorry for the screenshot ... Google Books doesn't allow to select text):
I unfortunately can't locate the bibliography section to back the claim, but a Greek text describing how the king of the Medes in Psittakos, Persia, erected a kitchen alongside a clear flame strikes me as a lot more precise than the vague claims seen on various oil/gas trivia sites. (The 120 CE date is weird though since that would be the Parthian Empire rather than the Medes.)
This article by Kaveh Farrokh is much more specific, and offers circa 550 BCE as the actual date:
The art of cooking is a long-standing tradition on the Iranian plateau. Ancient Persia is believed to be the first known region to develop piping technology for streaming natural gas to kitchens in the 550s BCE or perhaps earlier. The ancient Greeks certainly knew of Persian cuisine, at least as outlined by Xenophon in his Cyropaedia. Aristotle (384-322 BCE) for example refers to Persian cuisine as thus: “In Media and the district of Psittakos in Persia there are fires burning, a small one in Media, but a big one in Psittakos, with a clear flame. So the Persian king built his kitchen nearby. Both are on level ground, not in high places. They can be seen night and day.”
With the note: As cited by James & Thorpe (1994, p. 302).
The quote appears to be from the de Mirabilibus Auscultationibus (paragraph 35), which is usually attributed to Aristotle. The claim Persia developed pipeline technology first strikes me as weird if not dubious, since the Chinese arguably did it before them. Plus, there would be no need for pipelines if you'd build a kitchen atop an ignited gas seep. So that bit might be the author getting ahead of themselves. But the Ancient Greek citation seems genuine.
On the next page of her book, Snodgrass puts forward that the first use of gas to cook in a way we'd recognize today to have occurred in England in 1802 indeed - but mentions Frederik Albert Windsor of Braunschweig rather than Zachaus Winzler.
The claim that Chinese used natural gas to cook brine in order to extract salt since ancient times seems rather documented. I admittedly stuck to cursory googling but I haven't located any sources that claimed or hinted that they used it for cooking. One site (can't recall which) justified they didn't (without citation) because it would have required modern pipelines. Which I took as meaning safety reasons got in the way. (But I might have read that on a gas distributor's website, so take it with a fistful of salt.)
At any rate, going by the above, it indeed seems plausible that gas for cooking first occurred in Persia in a "brute" form (i.e. gas seeps), and in England in a modern form (i.e. gas pipelines). Who was involved in the latter isn't clear to me, but FWIW I'm tempted to trust the Smithsonian's Winzler claim over the Braunschweig claim mentioned above.