The issue regarding cape movement in all versions above and including 1.13 is a visual bug only. It does not seem to affect gameplay but it could do make playing with a cape very odd. I uploaded a video explaining this issue and pointing out the differences of how a normal cape is supposed to work, and how capes behave in updates after 1.13.
(I could not upload the vid on here because it said that the file was too big)
Link to video: https://youtu.be/4bnGkC2S3PU

The video does not do justice to the bug, watching it full screen might help.