This is clear and well-written, and makes sense to me. I don't think any of it conflicts with my statement, though (if you mean to correct rather than expand upon). My original statement is just a more general version of your more detailed divisions: in each case, "should" argues for a course of action, given an objective. The objective is often implicit, and sometimes you must infer or guess it.

"You shouldn't steal those cookies [...if you want to be moral]." More formally stated, perhaps something like: "not doing this will be morally correct; do not do it if you want to be a moral person."

"You should do X [...if you want to have fun]." More formally: "Doing X will be fun; do it if fun is desired."