Disclaimer: I'm in no way an expert in cymbals or recording (yay! guess that makes me fully qualified to provide answers on the Internet) but here's $0.02 from my experience.
This cymbal is not only short but also fairly quiet. I'd assume the drummer plays the kit more-or-less in the same way he does in the video. The assumption is that the song has been practiced thoroughly and there's not much point playing it differently on a video vs playing live or recording. The cymbals seem to be hit pretty hard so they most-likely do ring if you're around them. This means that the cymbals are set to a fairly low value in the mix, and only the attack is audible. Sounds almost as if the cymbals are picked by a mic that's a bit remote or even used for another drum. It could be an overhead mic on top of which the rest of the drums (using separate mics) are added with more volume. You might notice that all the cymbals in the mix are rather low - there's no hi-hat or ride at all, for example. The overall drum sound is pretty dry and there's just strumming of the guitars instead of the hi-hat beat.
There are other techniques if you want to get the similar sound live, though. People mostly tape the cymbals at the bottom or at the top. There are also magnets that can be attached. This dampens the cymbals somewhat and shortens the sustain. One can also adjust the technique and play with stick tips only, etc. But that will never give this much force with such a low volume unless the cymbal is wrapped in tape like a suitcase at an airport.
And the third option, naturally, is the characteristics of the cymbal itself. As suggested in other answers, splash cymbals will be high-pitched and have a short sustain simply due to their size. The cymbals in the video, i.e. at 1.14", seem rather small to me, apart from the ride. Definitely not 18"(!) but more like 16" or even smaller. Just compare them to the size of the snare. Meaning that they ring out decently in live mode but are only adjusted in the mix.