The photo looks right in an exported folder, but the little wave before the pose has disappeared. Before deleting the short video beside it as clutter, check whether the two files belong to one Live Photo.
Photos on Mac exports an unmodified Live Photo as a still image and a video. A file viewer may show them separately rather than reproduce the combined experience you see in Photos. Keeping only the picture is therefore a different choice from keeping the whole moment.
Decide what you are handing over
For a print, a still-image export may be all you need. For an archive, keep the original pair together, with a consistent naming scheme. Do not assume every receiving app will automatically recognise or recombine the pair.
For someone who simply wants to watch the movement, an ordinary video is easier to explain. On iPhone, open the Live Photo in Photos, use the More menu and choose Save as Video. Apple says this creates a video while retaining the Live Photo. Menu placement can vary with the Photos version.
Review more than the thumbnail
Try one item before a large export. Open the still image and play the video from the destination folder. Listen as well as watch: background speech or movement may reveal something that the selected still frame does not.
Ask the recipient to try the delivered file in their intended app. Keep the originals until that check is complete. A correct thumbnail confirms the picture, not that motion and sound survived the transfer.