When using the video player module it shows only the link with the video url.
I supposed it was the privacy but also changing to "Anyone" it won't work.
It was a caching issue. WordPress caches the video player on the first hit, so if the access settings for the video was wrong, then it won’t display the video player even though you fix the video access settings afterwards.
It can be fixed by going directly to the database (if you’re not sure what you’re doing, please remember to backup first!!!!!). I use phpMyAdmin for this.
Find the table named “wp_postmeta” (in my case I have prefixed my tables for security measures, so my table name is actually a bit longer)
click on “SQL”
Insert the following select statement:
SELECT * FROM wp_postmeta WHERE meta_key like "%_oembed%"
This will bring up all the cached embedded players. The meta_value field contains the player, so it should be quite easy to find the video that is not working and simply deleting the row (don’t worry, WordPress will generate a new entry the next time you visit the page and your player will be inserted correctly!).