Hi, Cindy - let me try to follow your steps and note what I think you've done in each one. Note that if you think one step is really not intuitive, we should discuss it on the infrared list and see if a different behavior would be a more intuitive way for most users.

1) I uploaded the image of a little Daisy here in London taken by my friend John with a NIR camera (blue filter 2007):

Yep.

2) I uploaded the image using "NDVI for BLUE filters" > Colorized and I got this:

Yep.

3) Then I choose RGB analysis > Run to get the NIR original image and then I select Grey analysis > Default colormap and get this:

Here, choosing RGB would get you the original if each field showed the value R, G, and B (in each of fields R, G, and B) when you pressed Run. Choosing Grey would get you just one channel - it depends on what's in the box next to it. By default, I think it shows (R-B)/(R+B). But did you press Run before pressing Default colormap? If not, it would just apply the default colormap to the old image, which might mean just one channel from that image, or the overall brightness across all three channels -- I'm not actually sure.

4) I wonder why the Daisy is showing red and apply the formula (R-B)/(R+B)*4 and without hitting Run I select Fastie colormap and get this:

Here again, without hitting Run, you're just applying the Fastie colormap to the same data left over from when you Ran the RGB equation, which we aren't sure of.

5) And when I hit Save, the image I download is this (for some mysterious reason):

Now I'm about to get on a plane, so I'll take a break, but if you've hit Save, we should see if that runs anything automatically before saving, because you'd think it should just save what's currently on the canvas. But as we're already in unknown territory, it's hard to say without more testing and a careful reading of the code.

So I wonder if we should not show the colorize button until you press "run"? Is there any situation where you'd want to colorize before pressing run? Also, what if you press run, colorize, then change the equation again, but forget to press run, and press colorize again?

I can't help much with the infragram.org process. It seems to be pretty easy to produce an image that doesn't have any obvious relationship to the original. If it helps any, one of your images is exactly what you would get if you computed NDVI from the original and applied the Fastie colormap:
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Unadulterated NDVI made in Fiji from John's original photo. This is the result you should get at infragram.org if you just ask for NDVI with the Fastie colormap. The NDVI values should be much higher for the healthy lawn plants.
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The NDVI values are too low because the camera did not have an appropriate custom white balance setting. It is very hard to correct this after the fact, but just as a test I swapped the green and blue channels which made all the blue values lower and increased the difference between blue and red for all the plant pixels. This produced a slightly orange version of your original photo:
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An arbitrary swap of the blue and green channels produced this image.
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That orange image looks more like a photo from a blue filtered infrared camera should look, although it should be much deeper orange. It produces an NDVI image with higher values than the original:
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NDVI image from the channel-swapped version of John's photo. NDVI values are higher and closer to where they should be.
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It's not necessary to swap the channels to produce this NDVI image. The same result is obtained if you use the green channel instead of blue as the visible data when NDVI is computed (red is used for NIR). I think Infragram.org allows you to do that.

To capture photos that make meaningful NDVI without additional manipulation, do a custom white balance while filling the frame with blue paper in the shade but under a blue sky.

Anybody know the name of that little daisy that has been blooming this month all through the lawns in London's Royal Parks?

I believe that if Cindy had pressed Fastie colormap instead of step 3, she would have gotten the same result as Chris's first image -- perhaps a slight variation as the Fastie colormap wasn't an exact copy of the one Chris has in Fiji. This is also what Cindy got after uploading the smaller version.