How do I list processes in htop by just their name, without the full path?

Apparently not. You can however go to Setup/DisplayItems to Highlight the basenames.

It doesn't only list the command like you want, but it atleast makes it a little more visible.

As an update, now apparently there is an option in Setup/DisplayItems Show Program Path that is checked by default. Just need to uncheck that so only the command name is shown.

To make htop display only nginx processes:

htop -p $(pgrep nginx | paste -s -d,)