Is a object that’s white with the lights on, still white although it looks different under a different light or in total darkness?
What makes an object white is its property to re-emit and scatter nearly all the frequencies (colours). During scattering, the directional information of the light that illuminated the object also changes; that is, the incoming light is scattered into a distribution of angles. To answer your question, an object's properties do not change if it is being illuminated by a different frequency or even if it is not illuminated at all.
While we are on the subject it is worth noting the difference between reflection and scattering. Unlike a scattering object, a reflective object preserves the incident angle. So a mirror is "white" in the sense that it re-emits nearly all the frequencies, but different from traditional white objects because it has definite scattering angles.
The answer depends on what you understand to be "a white object". Typically, we think an object is white if it looks white with normal illumination. Since the object doesn't change when you use a red light, or when you turn off the lights, it still is a white object. You just see it red or gray (or don't see it at all) at the moment.