I find it ironically amusing to see people who call SF "Sci-Fi" arguing over what hard SF really is.
Star Trek is what it isn't; Star Trek, in fact, is arguably Sci-Fi. It certainly does
not get the science right, or even try (and I've never forgiven them for forgetting about the speed of light in
Generations). Star Trek is an adventure story, with enough (usually imaginary) science pasted on to make it happen. Not, of course, that the science doesn't sometimes catch up with the fiction -- I can pull a small device out of my pocket, flip it open, and say the name of who I want to talk to, and it connects me. It's called a cell phone. But anyway ....
A simple definition of hard SF is that the story could not exist without the science. Not handwaving and technobabble, either, like the explanations for warp drives and transporters, but good, solid, real-life science. Usually the science comes first, and the story grows out of that, rather than the "science" being cooked up to justify whatever story the author wants to tell. The actual hardness can vary, but the real, or plausibly extrapolated from real, science has to be central to the story.
Hal Clement is one of the acknowledged masters, yes. Robert Forward is another. When he's not writing Star Wars books, Timothy Zahn has done some excellent hard SF (I personally like Spinneret, and would pay hardcover prices to get it as an ebook). Some of Larry Niven's books and stories are hard SF, though my extreme personal dislike for the man (we have history) tends to color my opinion of anything he writes. Many authors of softer stuff have also written one or more hard SF novels.