I've noticed that across videogames and anime that focus on fighting and action, the word "buster" is frequently applied as the name of a super powerful weapon. For example, Gunbusters from Toppu o Nerae, the Buster Sword from FF7, Megaman's Buster arm gun, Trunks of Dragon Ball Z's Buster Cannon. It carries this implication that the weapon is awesomely powerful. Clearly it is a word borrowed from and it has similar uses in different forms of English, but nowhere near to the same extent. Buster is like a go to word for these kinds of weapons in Japanese action shows and games, but it's rarely used in English for that specific purpose or with that much frequency. I'm wondering if anyone has any insight as to the origin of the use of the word Buster as it's used in this context, and any explanation as to how it might have become so commonplace compared to its English counterpart.
There's a number of words in Japanese that come from English, but not via the route expected. I suspect that バスター might be one such word.
Poking around in Kotobank's page for バスター, we see that this is probably the same word that arose from baseball, as a corruption or shift from the original English phrase bastard bunt, in reference to a play where the batter stands as if to bunt the ball, and then once the pitch is thrown, they shift to a regular batting stance in an effort to hit the ball with real force (as mentioned, for example, in this thread about Little League). See also the Japanese Wikipedia article on バスター.
If this is the same バスター that was used for these other weapons in manga and anime, it might have originally arisen as a play on the perceived littleness of the weapon or character, in contrast to the sizable wallop of the actual shot or strike: presenting as a small thing (like a bunt), but actually hitting harder than that (as an actual full baseball swing would do).
PS: As @jogloran notes in the comment, the specific phrasing バスターソード is most likely from bastard sword as a borrowing in toto (as a whole term), rather than an application of バスター from baseball. In the specific sword sense, the bastard portion was originally in reference to the way the blade length was roughly in between the single-handed shorter swords in common use and the double-handed greatswords of the time.
バスター is derived from Buster, which means "one that destroys, eliminates". It has the same use in English, like bunker buster, bronco buster, dustbuster, Myth Busters, etc. Some examples that are known in Japan would be the movie Ghostbusters and the pro wrestling move Brain Buster.
It is likely that the meaning "one that destroys" got simplified to "destroyer/destructive", and from there was widely applied. Once it catches on in one piece of work, it's easy to imagine people used it in their works referring back to another anime, as opposed to researching the original meaning.
A few notes in regard to your specific examples...
Gunbuster: Gunbuster is made up of two machines: Buster Machine #1 and #2. So the names of the those machines make sense. There was no info on the naming of the Gunbuster itself, though robot anime tend to favor the sound of the name over the meaning, so I wouldn't think too hard about that.
Buster sword: It could be a mispelling of bastard bword. When it comes to foreign words, sometimes mistakes go unnoticed (ベット when meaning ベッド), and this misspelling fell through the cracks. This was pre-wikipedia era, so it's not as easy as it is now to cross-reference.
However, this more than not to be likely intentional, since technically the sword Cloud uses cannot be classified as a bastard sword (= between mid and long size, double edged). It could be they just meant it to be a "sword that destroys".
Added in some references, though I don't think any are conclusive.
Buster entry on Nico Nico Video
Buster etymology article (probably the most conclusive I could find)
Buster Gundam Wikipedia article
Article on the meaning of team names of the Hypnosis Mic series