I really dislike the 6 traditional stats for many reasons, and this is one of them.
The chronicles of darkness games have a nicer stat system, in my opinion. It's 3x3. One axis is Power - Finesse - Resist, and the other is Physical - Mental - Social. They have names (strength, dexterity, stamina are the physical ones, for example), but this is the underlying concept.
Demanding people's attention is Social Power. Being subtle is Social Finesse. And keeping cool is Social Resist. Now it's possible to make a character that is The Center of Attention who isn't subtle, or someone who cannot be spooked but also isn't very good at talking to people.
If I was going to do some hacking to D&D, I would probably rip charisma out entirely. It's half-baked and its implementation introduces a lot of un-fun problems.