Overriding field.html.twig in my module
I managed to track down the place in core where the code for this specific theme is located, it was in core/includes/theme.inc and not in the field or system module where I previously searched. Copying from there did work in the end, using render element in hook_theme() is correct, but I was missing an important preprocess function. 

I defined the following function and called the original preprocess function from there:

function template_preprocess_field_image(&amp;$vars, $hook) {
    template_preprocess_field($vars, $hook);
}