Do not show picture when video available Got a content type with some text, and a video field and an image field (and some other stuff, like entity references between nodes, but not relevant for this question).

When displaying nodes for this content type, I don't want to display the image, if there is a video available.
So the image displays if the video field is empty.
If a video was added to the node, the image should not display.

Can only be done by modifying the node template I assume?
So I would need to copy the node.tpl.php  and rename it to node--contenttype.tpl.php   (with contenttype being the name of the content type)

But the PHP code in there just prints then entire $content  so I cannot interfere with the displaying of the image separately.  

Is there another customisation (the one producing the html for the node probably?) where I can wrap some "if" statement around the image field that check for presence of content in the video field?
You can use conditional fields module for this.
https://www.drupal.org/project/conditional_fields
and set the  dependency for your field. See the screenshot it will help you.