I would make the wild generalization that animated GIFs are the most common method to share animated snippets of UI work.
GIFs are extremely common in Behance case studies, and a ton of work showcased on Dribbble are animated GIFs. Not to mention they're are typically smaller file size than videos, and there are plenty of free tools on the web to convert video to GIF.
As for an all-in-one designing / prototyping / sharing tool, I would recommend Flinto for Mac. The experience is quite similar to Sketch (you can even import your Sketch files), it has incredible transition and behavior animation capabilities, and you can export a recording of your prototype to a video or GIF all from within the app.
There are plenty of designing/prototyping apps out there. Some popular ones would include InVision Studio and Framer X, but I cannot speak to their exporting features. I would just start with the free trials and see which you like best.