I think the qualifiers are there because the question of who came up with the first "stored program computer" is a matter of some notorious controversy - kind of unrelated to the first computer program. Knuth would have known about this.
"Stored program computer" refers to a computer where the program would be stored within the persistent memory of the computer itself, rather than elsewhere. It seems hard today to imagine such a thing, but think about punch cards or programs you have to enter in via adjusting switches or plugboards. The point of a stored program computer is that program running on the computer could, in theory, change the program itself (yes, "code is data" predates lisp, sorry).
According to Jean Bartik[0], Mauchly and Eckert's EDVAC was already a stored program computer, and thus implying that one of them must have written the first stored program. Jeremy Norman has the actual document where this happens[1], months before Von Neumann's publication. There's some evidence that some of the ideas in Eckert's paper were from an even earlier paper from John Atanasoff, the source of another bitter fight. In any case Bartik has a very poor opinion of Von Neumann, who apparently sat in on EDVAC meetings and wrote notes on the EDVAC without crediting Mauchly and Eckert, then allowed those to be distributed widely in his name, which in turn invalidated Eckert And Mauchly's patent application and thus also their claim to the invention in the public eye.
Another interview with Eckert[2] gets into the details of these allegations: That Von Neumann's motivations were that things should became public domain was because he specifically couldn't sell those ideas at IBM (for whom he was consulting) unless they were patent free.
Both of these are from biased parties of course so take their interpretations with a grain of salt, but that Von Neumann took ideas from EDVAC and summarized them in more formal language and did not credit the others seems pretty much settled fact based on other sources too. Von Neumann was already a well recognized and brilliant researcher in his won right, but reportedly he was also wracked with self doubt, so who knows, perhaps that could have been a motivation too. All of these were very ambitious people.
Curiously that interview also has a reference to another person claiming that a lot of papers attributed solely to Von Neumann were actually coauthored. Really gross stuff, though it's just an offhand comment without substantiation, unlike the other stuff here.
In any case all this reinforces my notion to be more suspicious of modern day "first" inventions attributed to a single person. Similarly to the telephone and lightbulb, there seems to have been a very active community of inventors (all quite important and impressive) who were dancing around related ideas and another might have gotten there eventually if one didn't. There are about 7-8 early computers, some of them within months of each other, that could all be considered early stored program computers. And the parties involved were exchanging ideas with each other too, one way or another. I'm not sure it makes sense to wring hands about which one was truly first.
[0] https://news.ycombinator.com/item?id=40333159
[1] https://www.historyofinformation.com/detail.php?id=639
[2] https://conservancy.umn.edu/server/api/core/bitstreams/7de38...