How to change the "Add new comment" message for custom comment form?
The Add new comment header is a part of theme layer. 

You can copy your theme's existing comment template (e.g. core/themes/classy/templates/field/field--comment.html.twig if you're using a theme that utilizes the base classy theme) over to your theme's template dir and rename it to match your specified field (e.g. field--field-my_custom_comment_machinename.html.twig). If you're not familiar with this, enable Twig template debugging.

Once you've create a copy of the template, you can then edit template header as you see fit.