With all due respect, the accepted answer has misleading statements.
First, things reported by customers are not bugs in the engineering
sence. They are defects that may or may not be bugs.
Even when we talk about TFS/Redmine and the like, the concept of a bug is very different from what is being expressed here.
According to ISTQB (and other qualification boards related to software quality) there is a difference between an Error, a Failure, and a Defect.
Let me explain each concept:
Error: is produced by a human being's actions (for now :) ) it's existence may or may not be detected and may or may not be classified as a software Defect
Failure: is how an Error can be detected while using the Application Under Test (i.e. error messages, data inconsistency, unexpected behavior, etc). A Failure may or may not be detected, but most often ends up being classified as a Defect.
Defect: is the documented description of a software Failure. It usually includes actual software behavior and expected software behavior. More and varied information can be included as well (i.e. software version, environment, etc)
Now what is a "Bug"??
"Bug", as a term, is a software testing slang that dates back to the late 40's.
Check this out:
http://www.computerhistory.org/tdih/September/9/
Taking all the aforementioned, we can say that the Work Items in TFS and Issues in Redmine both have a "Bug" category that refer to work items that allow Defect management actions.
In conclusion, whether or not the unexpected behavior was found by a customer does not change the fact that a "Defect" is a "Defect", and therefore a "Bug" is a "Bug".