GIF is an old and very inefficient format. It dates from 1987, compresses poorly, and can store only 256 colours per frame. A few seconds of smooth animation can easily run to twenty megabytes as a GIF, while the same clip as H.264 video might be under a megabyte and look better. At the scale X operates, serving real GIFs would be an enormous and pointless bandwidth cost.
So X does not. When anyone attaches a GIF to a post, it is converted to a silent MP4 immediately and permanently, and the interface simply overlays a small GIF badge so the thing still reads as a GIF to viewers. From that point there is no GIF anywhere in the system. The original was discarded after conversion, which is why saving one gives you a video and why no tool can retrieve a GIF that no longer exists.
Rebuilding one is possible and is what the GIF options here do, but it is genuinely lossy in a way worth understanding before you choose it. Reducing full-colour video to 256 colours per frame introduces visible banding in gradients and skies, and the resulting file is usually several times larger than the MP4 it came from. Those are properties of the GIF format itself, not limitations of the conversion — any tool doing this faces the same trade.
For most purposes the MP4 is simply the better file, and it is offered first here for that reason. Modern chat applications, Slack, Discord and most forums accept short silent MP4s and loop them exactly like a GIF. Rebuild an actual GIF when you specifically need one — an old forum, an email signature, a system that rejects video uploads — and take the MP4 the rest of the time.