There's no download button anywhere on X — and 'GIFs' aren't even GIFs, they're short MP4 loops served in HLS fragments, so right-click-save doesn't work either. A downloader reassembles everything for you automatically.
Open the tweet, tap Share → 'Copy link to post'. Old twitter.com links and new x.com links both work identically.
Open the Twitter/X Downloader, paste the link, press Get video. You'll see the video or photo preview.
For videos pick a quality: top option is the best MP4 X serves. Tweets with up to four photos photo-download as one ZIP at full size — X stores images up to 4096px.
Protected tweets are visible only to approved followers, and that's a hard boundary: they can't be fetched publicly, and honestly, any tool that claims otherwise wants your login for the wrong reasons.