The correct one is "make a mistake". The verb "commit" is usually reserved for something more serious, such as "commit a crime". Typical, yes, but it's the only one I can think of at the moment.
That's an interesting theory, but I doubt whether most errors are intentional.
Here's another differentiation:
In applied linguists, an "error" as it applies in "Error Analysis" is a mistake that learners make because they do not have the linguistic competence to know better. A "mistake" is an error they make when they do know the correct form, but are under pressure, or aren't concentrating.
If I wrote "a understanding" or "What's it's meaning?", then I have made a "mistake", because I know that the correct forms are "an understanding" and "What's its meaning"
If a ESL student wrote that, it would sometimes be impossible to know if it was an error or a mistake - ie. do they know it's wrong?