Based on the provided information, it appears that customizing task types beyond the default options (Email, Call, To-do) isn’t supported. HubSpot currently allows the selection of only these predefined task types.
These requests submitted to the HubSpot Ideas section of the community are reviewed by the HubSpot product team, based on their popularity and the assumed demand. I’d recommend commenting and upvoting.
In the thread, the product team also has a form where you can submit a survey.
You can also help other HubSpot users find this request more easily (and drive traction) by accepting my reply as a solution. I’d appreciate it, too.