By using a filter of the type URL parameter as such:
You would then add the filter value to the URL as such:
https://my-host/api/lsw_tasks/task/v1?task_name=thenametolookfor
To return the type of a record, you would add it as a column. If the type name is not on the same relation as the original record, you can join the relations (as you can see in my screenshot, where I joined state to task because the state name is not on the task relation).
