Don't blame Greg, Phil. You've got to read the passage you quoted in context. Early in the article we read "Herschel's first
comet discovery came on the night of 1st August, 1786".
But I don't think NGC 253 is correct either. There are webpages saying so, but if you look at the records there is a enough ambiguity that any one of around 10 or so objects might have actually been the third discovery.
http://cometography.com/biographies/herschelc.html
I believe the above article is so definite about NGC 253 being the third because it's easier to tell (make up) the story in this way. First this happened on this date, second this on that date and third another thing on another definite date. Bang, bang, bang. Now that sounds really convincing, and makes the author sound like they know their stuff. But if they honestly said that there were these dozen or so early discoveries in 1783, a few of which we have exact dates for, but most of which we don't know exactly when, and some of those might not really have happened at all, or might have happened later or were due to someone else, and there is some conflicting data and ambiguities in the historical records and so on; that sort of thing is much harder to narrate.
www.seds.org has the facts on C. Herschel along with useful references.