Somehow urls were broken, maybe from too much legacy mode switching... Neither friendly URL desactivation or image regeneration worked.
Managed to "workaround" this by copying an old backup on wamp, in order to have safe_mode off... Then, this "move images" button was here, and totally functionnal, with image regeneration and all I needed (I don't know why I haven't thought about it earlier...).
Thanks for your support !