I am not sure about these labels. One clue of the misleading is that you needed to put a note on "Completed with failures". The need for a note indicates that the label is not intrinsically clear to users.
Why not simplify the concept here:
- not started
- Started
- In processing
- 4/5 completed
- 5/5 completed
I would use a visual code too (colors like red, yellow and green, for example)
moreover, it's unclear what's the difference in status between "started" and "in processing" (me as a new user do not understand, if a task is started it should also be in process. Not being so I create a mental model of the system in which there is a middle state of a task started but somehow still not in process.. so what's this status?). Most probably there is a reason for you to create two different status "status" and "in processing", maybe you should review this label too?
The idea of a task completed with failures somehow does not sound right. In my mental model if I have a task, and 4 on 5 subtasks failed, that main task is not completed, so a "completed with failures" seems to me a strange communication. What my next actions should be? Be happy about it because it is "completed" or being worried because one subtask failed?
In summary (taking in consideration I do not know anything about the system) I would use these labels:
- To do
- 0/5 completed
...
- 4/5 completed
- Done