Is there a reason users would want to know if the error is a user error or a system error? How would this help them, as long as they know what to do in each instance (this would be an explanation of how to resolve the issue if it's a user error or instructions to contact support if it's a system error with unknown cause)?
Short answer : yes.
Long answer: Giving the user a clear indication of the origin of the error provides clarity.
Telling them what to do after the error is not enough, the user is still missing the comfort / knowledge of knowing the origin the error.
There has been an error. Please contact customer support to advance.
- Oh no, did I mess something up? Is my account locked? Did I break something with the invoices? Should I try and find what I broke before calling the customer support? I really don't like dealing with them.
There has been a system error. Please contact customer support to advance.
- Ugh again? This service breaks all the time. I guess I'll have to call customer support.
I've found in user testing that users will often blame themselves for system errors and think that they broke the system. Letting users know that the error was systemic can assure them that it was beyond their control; kind assurance and next steps help support a better user experience than keeping things vague.