Well, you’re almost there! If you want to include a snippet in the textarea, you’ll have to create a custom tag, called ‘snippet’ or something. Texnine explains it here:
https://forum.getkirby.com/t/embedding-snippets/900/4
What is ‘the right place’ in this case? 
If you want full control, then you can try Tim Oetting’s Kirby Builder plugin, which gives you ‘modular blocks’ to create pages with. One of these blocks can be your form. But maybe that’s a too hightech solution to your problem :).