Dear Mareike,
if you are creating JSON with the template source code, did you define the template content within the format templates (e.g.
<p>$CMS_VALUE(#content)$</p>
für the p format-template)?
As long as the JSON is created with the use of the template source code (regardless, if #global.json.sectionTemplateRendering is activated or not), the content of a CMS_INPUT_DOM is created as HTML defined by the appropriate format templates used within the editor.
It should look like this:
{"fsType":"CMS_INPUT_DOM","name":"st_input_dom","value":"<p>Absatz<\/p><p><b>Fett<\/b><\/p><p>\"<i>Kursiv<\/i>\"<\/p><p><a href=\"https:\/\/heise.de\">Heise<\/a><\/p>"}
If you want to have exactly this, but it doesn't work, then please post the source code you use in the format templates (especially in the "p" template) and the JSON that is generated.
Best regards
Holger