I had a closer look at the generated HTML and found that both wiki links create the same HTML:
<img src="../wiki_uploads/5/51/My_picture.png" />
That relative path is wrong for a subpage as there's one path level missing. So I suppose the problem has something to do with my path settings.
I found the solution. I had to change the upload path so it includes the article path like so:
$wgUploadPath = "{$wgArticlePath}/../wiki_uploads";
Thanks again for your help Frank