I agree that:
(1) There is a difference between science fiction and sci-fi, and
(2) I like both of them. "Sci-fi" is not a derogatory term, just a term describing a different kind of storytelling.
Or maybe I should put it this way: There is stuff out there that I call "sci-fi" and stuff that I call "science fiction". Whether those are the proper terms everyone uses to describe those things, I don't know. What I do know is that there are in fact different types of things that qualify as "fictional stories told with a fantastical scientific slant", and I think two terms are required to differentiate them.
I grew up in the reading the literary science fiction of Asimov, Bradbury, Bova, Clarke, Dick, Ellison, Niven, Robinson, etc, and find those sort of stories to be very different than the way stories are told on Star Trek. I chose to call that former stuff "science fiction" and the latter "sci-fi". I love both things; one is not better than the other. However, both things are different. The latter having a "sciency or spacey" backdrop does not make it just like the former.
EDIT TO ADD:
I will say that a few Star Trek episodes have included themes that I would call Science Fiction.