What movies/TV shows qualify as "hard" science fiction?

Well, IMdb says it made $5,000,000 which wasn’t chump change back then. It wasn’t a cosmic shoot em up, to be sure, but certainly fits my definition of hard SF.
The article you refer to was published in Astounding about the time the movie came out. They worked hard on getting everything right.

As for the magazine, I suspect you are referring to Weird Tales. My Day Checklist shows there was a Weird Stories magazine, but it was British and only lasted two issues. Weird Tales published Lovecraft and Robert E. Howard, but it was horror and fantasy mostly - I can’t think of a lot of science fiction that ever ran there.
I’m sure “Weird” was used by the marketing people to try to space up the not very spicy movie. Or maybe in 1950 that going to the moon stuff was considered weird by the masses. Destination Moon is about as far from weird as you can possibly get.

Your mom is right.
Science fiction was not published in Weird Tales - except perhaps in rare instances, none of which I can think of at the moment.
I don’t own any copy of the original magazine, but I have an anthology from 1965 with some stories. There is a non-Conan Robert E. Howard story. The introduction quotes August Derleth as saying that a small but devoted following keeps Howard’s name alive. That has sure changed!
The rest of the stories are fantasies also.
There was a revival of Weird Tales from 1988 or so to 1991. Some fantasy, a lot of horror, no SF I can recall. (I have a lot of that run.)

I don’t think that there’s anyone at all who would disagree with Destination Moon being “hard SF”, by anyone’s definition.