Here's the verdict: Looking at the photos in both Firefox and Safari on both computers at the same time and with the desktop in sRGB emulation - there is a difference on the laptop but perhaps not huge like it would be if the flower was red.
On the desktop-no difference between the three in Firefox but there is a slight difference between Safari and Firefox- the colors are very slightly more saturated in Safari. It is my understanding that Firefox has better color management than Safari but not sure about that - I do see a small difference though.
On the laptop-in Firefox - sRGB is more saturated (but not to the point that I would say OMG it is just more saturated and if I wasn't doing a comparison I might think it was fine). There are some pinks or reds in the background that show more color too. Those pinks/reds are not terribly saturated but that background color does not show up on the desktop with as much strength. The flower listed as aRGB is less saturated than the previous one but more saturated than the final one, ProPhoto. The ProPhoto looks closest to the photos on the desktop. In Safari on the laptop not a huge difference between the two browsers.
I went to my friend's house earlier so I didn't have a chance to see the sunflowers on his computer. On Flickr using his iMac all of the photos I posted with the tomatoes look pretty much the same. (I think the color on his machine may be slightly more saturated than on my desktop and the NEC but the color of the tomatoes look normal just a bit more pop to the color. His iMac has not been calibrated with a device and I believe he said he has not calibrated in a while.) On my laptop viewing the same photos on Flickr, the ones without the embedded profile look less saturated almost normal but a little too desaturated. I also took those same photos on a thumb drive and we opened them in Preview on his desktop. They all look the same. On my laptop opening them in Preview, those with the embedded profile are very over saturated and those without are desaturated - exactly as they appear on Flickr from my laptop.
I wrote my Flickr friend to see whether she was still using her NEC and still having the same problem. She said she is using Vista as her operating system. This is her partial reply:
"Yes, I am still using the NEC monitor but have given up on using the wide gamut option. I use spectraview to calibrate and use the sRGB emulation mode with the white point set at D65. If I save out of photoshop with the save for web option, the colors seem to be fine, or at least I like them. I was using the wide gamut option for a while, and then converting to sRGB for flickr, but decided that it was too much of a hassle. The profile in photoshop is Adobe RGB.
I myself am having lots of trouble with photoshop because I upgraded to CS6 without realizing that it doesn't work well on Vista. One day (soon) I will upgrade the operating system, and hope the problems will go away."
So apparently she is still having a problem as well but has found a work around that satisfies her. I don't particularly want to have to edit in sRGB and go through the steps she is using since I am using Lightroom primarily and by doing that, as has been discussed, I would then be editing to satisfy the laptop, right?