Change the display name of a default field in Contact Form
AFAICT each contact category has its own form ID, which means you can change it with hook_form_alter() in your theme or a custom module. Here's a tutorial about  how to implement it, although there are other resources you can find: Learn how to change an existing Drupal form. 

If you wanted to do it without coding then creating a new field and adjusting form and display for each contact category is the way to go.

As Hudri mentioned in a comment you could use Webform for a lot more control, including saving the responses in the database. If you don't need to keep record of the correspondence than Webform is probably an overkill.