I think your first paragraph more or less states the difference: (1) is more common and (2) can be used in a more abstract sense as in "I have the feeling that....". Another point may be that usually implies an animate "feeler".
* (literally) The feeling of the interview was good.
* I have the feeling that the interview went well.
Both are examples of "abstract" usage and basically say the same thing. cannot be used here.
To me, is used for describing the feeling of (e.g. surface of) something.
* / This cloth has rough texture.
In this sense, texture/touch may be closer to . Here may be less common but acceptable.