I've seen you paste this a few times now and find myself disagreeing with parts of your framing.
First, I don't know any mathematician personally who makes such a clear linguistic distinction between 'exercise' and 'problem'. Once you get to university-level mathematics, many exercises are problems in your sense but they still tend to be called exercises or something similar. If you insist on this terminological divide, I doubt most people will understand you.
Secondly, there is the matter of an exercise's pedagogical purpose. Is it to sharpen general problem solving skills or to enlighten the student on a conceptual level? This goes beyond difficulty. It's a false dichotomy when stated so simply, but there is still something there. Many IMO-style problems are conceptually barren but still very tricky to solve. Conversely, some of my most enlightening learning experiences were solving guided sequences of exercises in a mathematical form of Socratic learning where none of the steps were individually too hard but still involving enough that they forced me to think and thus develop some insight on my own. (This approach can also fail. Silverman's otherwise excellent book Rational Points on Elliptic Curves has a guided proof of Bezout's theorem in the appendix that is just too atomized to engender much understanding.)
Those are well-framed responses. Thanks. Khan Academy elicits my response because SO FAR, among the online exercises I have tried there, the "enlighten the student on a conceptual level" hasn't happened as much as just the habit-clinching drill. As Lang wrote (as quoted in the grandparent post), "Of course, some rote drilling is necessary. The problem is how to strike a balance."
The "mathematical form of Socratic learning where none of the steps were individually too hard but still involving enough that they forced me to think and thus develop some insight on my own" is what I attempt to provide in my live, face-to-face mathematics classes. I'm not worried about Khan Academy reducing the market for those classes (and in fact encourage current and prospective students to try out Khan Academy) because providing that sort of instruction is very hard to automate. As your example of Silverman's book points out, it is more of an art than a settled science to decide just how many steps to show with Socratic guidance, not to mention that different learners need different steps drawn out for them.
First, I don't know any mathematician personally who makes such a clear linguistic distinction between 'exercise' and 'problem'. Once you get to university-level mathematics, many exercises are problems in your sense but they still tend to be called exercises or something similar. If you insist on this terminological divide, I doubt most people will understand you.
Secondly, there is the matter of an exercise's pedagogical purpose. Is it to sharpen general problem solving skills or to enlighten the student on a conceptual level? This goes beyond difficulty. It's a false dichotomy when stated so simply, but there is still something there. Many IMO-style problems are conceptually barren but still very tricky to solve. Conversely, some of my most enlightening learning experiences were solving guided sequences of exercises in a mathematical form of Socratic learning where none of the steps were individually too hard but still involving enough that they forced me to think and thus develop some insight on my own. (This approach can also fail. Silverman's otherwise excellent book Rational Points on Elliptic Curves has a guided proof of Bezout's theorem in the appendix that is just too atomized to engender much understanding.)