Shares, likes and comments all count towards the total count on the Facebook Button. NOTE: Nudge currently only reports on the Likes and Shares in the dashboard.
Facebook will only count a share if the dialog box is not removed when posting. If the digalog box is removed, it will be classed as a "regular post" and therefor the share count for that link will not increase.
The image displayed in the share dialog can be removed as long as the link text and link stays present. This will class it as a "share post" still and not a "regular post".
If a url is shared and people "like" the "share post", the button count for that url will increase. eg 1 share + 20 likes = 21 on the FB button count.
If a url is shared, people like it and the share is later removed, the button count will only retain the share not the likes. eg If 1 share with 20 likes is deleted, the FB button count would now only show 1. The likes would be lost.
If a url is shared privately in a chat message, that will not count as a share.