Do you mean that the calibrated result is different depending on whether you calibrate with the ambient light on and off? That is, if you create two profiles in these two conditions and then compare them (either by observing the correction curves in the software or by switching between them quickly while looking at a test photo), they happen to be different?

If so, this is a measurement problem: probably the ambient light leaks into the sensor. The sensor should be placed right against the screen surface without gaps (but without pressing it so as not to distort the image/colours). The sensor should measure the light from the monitor, and it doesn't depend on the ambient light.

If you can't eliminate the gaps, you should do the calibration in darkness. If you say that your result is very different to the built-in sRGB, you should try it just to see if it has an effect. (Eizo's factory settings can be trusted, in general).

If, on the other hand, you are saying that the same calibration looks different with different lighting, this is perception problem. Objectively, the image on the screen is the same.

(Some monitors have an ambient light sensor and a function to self-adjust, but usually they are supposed to adjust only the brightness/contrast, without affecting colours. But in a professional environment, this feature is not used, and generally shouldn't be relied upon).

You should certainly not try to tweak the profile "to taste". What's the point of calibration then? Instead, you should adjust the environment. This doesn't only concern the ambient environment. Strive to use neutral grey colours for the windows and other GUI elements; this way your eyes will adjust better to the on-screen environment (which we know is stable after calibration). Trust more what you see on the screen than what you see around you. (Once you confirm the calibration works properly, of course).