Although the topic has been discussed a while ago, I thought to share my thoughts on this, in particular because what I learned during a colour management workshop contradicts with the conclusion that you cannot work around different lighting circumstances with a colour checker (my apologies if I misinterpreted the remark made). It is the colour checker's very intention is to get consistent colors across various light circumstances.
Perception of color is very subjective. What you see may differ from what someone else sees. It depends on the context, what our brains expect to see and illumination. Although the human visual system is able to adjust to changes in illumination and preserve the appearance of a color, it can also be easily 'tricked" seeing a totally different color than the true color.
The "only" thing we can do in photography is to capture “true colours” and create consistency among our devices (camera, display, printer, projector) by calibrating the equipment and by aligning the settings of each device. The purpose of the color checker is to compare the colors captured in a specific lighting setting and compare those colors with the true colors as defined by the manufacturer of the color checker. So ideally, you make one photo of every scene with the color checker and create a profile for that situation. Obviously this is not very practical, especially since there are other factors that determines how other people will experience your photo (their set-up, their lighting context, their expectations). So pre-creating camera color profiles like Ann does, is a more practical solution.
This is how it works with x-rite and Lightroom:
1. download and install color checker passport software, including plug-in for Lightroom.
2. Make references photo with color checker. E.g. for each camera: Shoot color checker in various circumstances: sunny day, cloudy day, rainy day, inside light room, inside artificial light, flash light. Or whatever circumstance that is relevant for you.
3. Export these reference photo(s) with preset for color checker (under file menu).
4. A DNG profile is generated and visible in the basic menu of development module.
Note: You can find profiles in user>library>application support>adobe>cameraRaw>cameraprofiles
Having said so, at the end of the day what is relevant is what you want to communicate with the photo or what emotion you like to trigger. To what extent "true colour" matters for achieving that, is up to you. You can create your own profiles, buy profiles, or get profiles for free. I like Ann's suggestion to start with using Adobe's profiles and then create your own.
PS 1: According to the teacher of the workshop, some checkers are more accurate than others. We did not have time to dive into what true color is and which device is better than another. I got the impression that he liked x-rite's color checker though.
PS 2: I do hope I correctly reflected what was said during the workshop. If not, experts at this forum will hopefully jump in.