With the exceptions of sollen which should be shall instead of should (which would be sollte) and - as chaero has mentioned in a comment - to have to which is more often müssen than sollen, your structure is ok, but it must not be taken 1:1 (mind my usage of must in the last sentence). Similarly, there are overlaps between mögen and may... especially when it comes to their true modal usage. So it is more like a loose guide.
German and English modal verbs are not the same. Not all the verbs you mentioned are modal verbs in English (like, want, have to, be allowed to). Also, the definitions of what exactly a modal verb is are different in both languages.
One thing to understand is that modal verbs in German do not exclusively express modes. In fact, können and mögen are mostly used as stand alone normal verbs.
- Ich kann Deutsch sprechen.
is not a modal expression, when you mean that you are able to do it.
is not a modal expression either.
As far as the categories go, I have recently read that grammarians distinguish between only 2 groups. One was obligation and the other one was (not sure though) potential.
So in order to categorize them you first have to single out their pure modal meaning.
Etymologically, the correspondence is like this:
- müssen - must
- sollen - shall
- wollen - will
- mögen - may
- dürfen - (nothing to my knowledge)
- können - can