Is the Xbox 360 power button "safe"?

On a powered-up PC, pressing the ACPI power button doesn't cut the power - it instructs the operating system to shut down cleanly, first saving your work, flushing caches etc.

Does the glowing touch-sensitive power button on the Xbox 360 operate similarly or does it just cut the power?

Many games warn you that they auto-save and not to turn off the console at certain times - are they only referring to cutting the mains power?

Actually it doesn't do anything amazing. It just simply cuts the power. Unlike a computer, the XBox doesn't have any major work that needs to be done when switching off. There is no data that needs to be saved, and there is no processes that really need to be quit cleanly.

The XBox was built with a different concept in mind then a PC, and therefore, just pressing the button, pulling the power or even using the control to shutdown the console, all do the same thing, cut the power. All important data on the XBox is saved at the point it needs to be saved, and it hardly if ever, keeps important data in memory.

From comments

At the time of writing the answer I forgot about the scenario of active downloads. If you have configured you XBox 360 to continue downloading if when off, the XBox will go into a sleep state when using the controller to switched off, with the exception of pulling the power.

I think it is safe.

I just observed the following: when the Xbox crashed and locked up, the first press of the power button made the characteristic beep, but did not turn it off. (the second press did so, presumably just cutting the power)

I take this (combined with the "background downloading" feature) as strong evidence that the power button, at least on its first press, does indeed act like a PC's ACPI power button, and asks the operating system nicely to shut down. The fact that it usually powers off within a fraction of a second just shows that the Xbox is doing a lot less than a typical PC.