A simplified version of the question in the title is "is lo-res color #1 red, or is it purple?" because that's the color that shows the most obvious variation. Color #1 was usually called "magenta".
In this video, of a IIe with an Apple-branded composite monitor, the bird on the credits page and the rightmost bird in the main menu are lo-res color #1. As you can see, it's purple. I wouldn't call it magenta but maybe some people would. To the left of the screen you can see a printed card shipped with the game, showing what I think is the intended color of that bird: red.
In this video you can see an Apple running the BASIC Kaleidoscope demo. As you can see, color 1 is red. The video description is in Japanese and Japan uses NTSC(-J). It says it's a II+.
In this video of Brick Out on an unknown machine and monitor, the background of the title screen is color #1. It's purple.
In this 1988 episode of The Computer Chronicles, showing a IIc Plus—a demo machine from Apple, operated by an Apple employee, with an Apple-branded monitor—the color in the upper left is #1. It's red. I don't think they're using an RGB card because it would have made a mess of the menu bar. The Video-7 cards had a color/mono mixed mode to avoid that, but this game (Math Blaster Plus!) doesn't seem to support that mode—at least the versions on Asimov don't.
Beagle Bros. posters show it as red, leaning toward the magenta, but nothing like the purples you see in those videos.
On the IIGS it's $d03, which is much more red than purple, and is called "deep red" in the control panel. The IIGS only does RGB and its simulated artifact colors aren't very accurate, but this is a vote in favor of red by someone at Apple.
Okay, well, tint dials can be set incorrectly. Although apparently the ones set incorrectly are all set to the same wrong value. What does theory say the color should be?
The video system outputs pixels (560 per line) at 4× the NTSC color carrier frequency, so repeating 4-pixel patterns get uniform colors. The colors 0001, 0010, 0100, 1000, 0111, 1110, 1101, 1011 should all have the same chroma phase mod 90°, and 0011, 0110, 1100, 1001 (the hi-res colors) should be offset from that by 45°. The harder question is: how much are they all rotated relative to the reference color burst? Or, to simplify again, what is the phase angle of color #1?
The Apple II Circuit Description by Winston Gayler, p.130, says that the colorburst signal is offset by 35ns (45°), which would mean that lo-res colors 1,2,4,8 would be at 0° phase (mod 90°) if there were no other delays. The color burst phase was adjustable by the "color trim" pot on the motherboard; I assume he's referring to the factory setting.
All pixels go through a flip-flop. Understanding the Apple II by Jim Sather, page 8-16, says
An interesting point about the picture flip-flop is this: its low-to-high propagation delay (time after 14M rises for pin 5 to go from low to high) is 13 nanoseconds typical, but its high-to-low propagation delay is 25 nanoseconds typical. [...] For example, a white dot of a normal TEXT pattern would be about 24 nanoseconds longer than the black dot of an inverted TEXT pattern.
The accompanying figures show this widening, and also a shift of the center of the pulse by, I assume, (13+25)/2 ≈ 19ns, or about 24.5°.
That makes sense. But I haven't found anyone but Sather who mentions a flip-flop delay. He doesn't even mention it in Understanding the Apple IIe, and the diagrams in that book don't show the widening, though they still show a delay. Gayler explicitly says that color 1 is at 0° (mod 90°). This presentation (by Ferdinand Meyer-Hermann?) has blurry photos of oscilloscope traces that appear to show that 1,2,4,8 have about 0° phase, or at least less than 24.5° (although wasn't the Apple's video output much closer to a square wave than that?).
A further complication is that there are two different NTSC RGB standards, and they are very different from each other. SMPTE 170M makes a big deal of specifying the linear-RGB-to-composite-signal encoding very precisely—quoting some conversion factors to six decimal places—and then makes it all meaningless by saying that you can use either of these sets of RGB primaries, which don't even agree to one decimal place. It makes no sense to me, but that's how it is, seemingly.
Let's try all four combinations of hue rotation and RGB primaries:
As you can see, color 1 is red if the angle is 0° and purple if the angle is 24.5°. But those can't both be right, can they?
I'm being a bit disingenuous because the story changes when you look at other colors. Color 8, especially, is very greenish with a 0° angle. This color is called brown, and looks brown in all of the videos where it's visible, which argues against the 0° shift. In the video of the IIc Plus, I can't get color 1 and the others to look right at the same time with any rotation angle and either set of primaries. The Japanese video is consistent with a 0° angle, but it could be they fiddled with the tint.
The SMPTE primaries seem a better fit than the NTSC 1953 primaries, but just to complicate things further, AppleWin issue #253 has a photo of a screen with a very bright red that seems a better fit to the latter. Also, of the four color-1 candidates, the 24.5° NTSC variant is the only one that I'd personally call magenta.
I'm seeking any information that might prevent me from ending up in a padded cell. The #1 thing I would like to have is a high-quality 50+ MHz recording of the raw video output including the color burst. Failing that, I'd like a second source for the alleged flip-flop delays, information on how many TVs and monitors of the time used NTSC 1953 RGB and how many used SMPTE "C", and as many photos/videos as possible of vintage equipment showing lo-res color bars. It's very hard to find examples online because most "Apple II" videos use emulators, and of the rest, most use monochrome monitors, or LCDs, or only show hi-res, or (almost always in fact) the image is overexposed.
My best theory right now is that the Apples that produce red are secretly PAL models, or using some other non-NTSC color generation method, despite evidence to the contrary. The videos of PAL Apples that I've seen usually show red, and the company that made that bird game is German. But here's a PAL Apple showing purple, and that's not the only hole in that theory.
$0d03(i.e. R = $d out of $f, G = 0, B = $3, equivalent to HTML #dd0033). Cf. the border colour values listed in 1000bit.it/support/manuali/apple/technotes/iigs/… . This doesn't answer your question because — amongst other reasons — I cannot prove that Apple's RGB choices are intended to be accurate rather than ideal.