Glass should have no color.
If you give a glass shader a color, you will get, as you pointed out in the question, "unrealistic" relfections. The reflections should indeed be untinted.
Here's an example:
What gives glass color mainly is the light that is absorbed as it goes through the surface.
A volume absorption node will give you a more realistic result, with "white reflections".
To understand when you should have colored highlights you can watch
Andrew Price's video on the principled shader (https://youtu.be/4H5W6C_Mbck?t=646). The important concept is that dielectric Materials (including glass) will not have colored highlights. Reflections on metallic materials will be tinted with the color of the material.