The error message still appears in a certain program, so it should be sufficient to tag it with that. Think about it: Would anybody ever follow error-message, because they're an expert in… error messages?
I personally don't know any expert in alerts or warnings, so I assume the same arguments are valid in this discussion.
You would differentiate between the two if there are different severities for the error/warning/alert/info.
http://msdn.microsoft.com/en-us/library/aa937483%28v=sql.80%29.aspx?ppud=4 has a gazillion levels, which can be useful to a user who wants to filter out severities below a given level.
You would lower the severity threshold when you want to get every last warning out; raise the threshold if you have to get something out quickly and don't have time to sweat the details.