Within a record, why am I not able to delete a certain type of activity?

This due to a permission setting by your account administrator. If you're attempting to delete an activity and not seeing the delete icon, (see below) this is related to a setting enabled that does now allow for deletion permissions.

If you believe this was done in error, please contact your account administrator, Crelate support will never change user security permissions. Full permissions will need to be granted (see below) for activity deletion permissions to be granted.