It does offer the option to let it do the color management when printing, but it doesn't help me to turn off color management in the printer
Hmm, I wonder why they did that differently in PS Elements? In Photoshop, if you select to have PS handle color management, all printer color choices under the Print Settings button (shown above in the image) are automatically disabled and grayed out. You couldn't turn them on even if you wanted to.
I set Rendering Intent to "Relative Colorimetric" (without knowing what it or any of the other choices means)
For Perceptual and Relative Colorimetric, intent tells the system how to handle out of gamut colors (colors that are outside the boundaries of the profile being used).
With Perceptual, colors beyond the profile borders are moved in to the nearest color that is in the profile, and all other colors already in the profile boundaries are also moved to attempt to maintain a relative hue/saturation distance from those being moved in from the outside.
Relative Colorimetric also moves out of gamut colors in to the nearest color within the profile, but colors already in gamut stay where they are.
Imagine it like an accordion. With Perceptual, as you squeeze the accordion, every fold moves inward. With Relative Colorimetric, the only folds that will move are those from the outside to a given point inward. The rest of the folds will be locked, as if they were made out of wood, or other stiff material.
Letting the printer do the color management, once with Automatic and my profile specified as the default profile of the printer, and once with my profile specified directly, gave results that were visibly different than those produced by iPhoto.
Which at least says that the two apps are handling color differently, but I can't say how from here. I'd practically have to be sitting at your computer. But it could come back to something I mentioned early on. And that's the possible issue that the printer is applying some kind of automatic color adjustments right at the printer, with no way to turn it off.
One way to maybe prove that is to make a simple test image. Put blocks of random colors on it, but nothing close to white, or darker than, oh, a 60% gray. Essentially pastel colors to somewhat dark. Print that. If the printer is applying unwanted contrast moves all on its own, your lighter colors will come out pushed towards white, and the darker colors towards black.
By the way, they claim that there is no Gamma in printer profiles. Can that be?
That is technically correct. For instance, a monitor profile contains a LUT (Look Up Table) that is loaded to the video card when a monitor profile is applied. The LUT controls white point color, gray balance and gamma. You'll notice when you start a monitor profile, the monitor's settings automatically shift. That's the LUT being disabled by the profiling software so the monitor can be profiled from an uncontrolled state.
A printer profile has no gamma. It doesn't need one and actually doesn't apply to that type of profile.