Are the Reds the oldest team in baseball?

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This article:

http://blogredmachine.com/2013/03/07...in-cincinnati/

was posted over on r/reds, and a conversation was started over the fact that the Red Stockings that were expelled from the NL in 1880, and the Red Stockings that formed as a charter member of the American Association (today's Reds, who eventually re-joined the NL) are not technically the same franchise, and thus the Braves and Cubs are older clubs. Basically, Wikipedia breaks it down like this:

- 1869, the Cincinnati Red Stockings become baseball's first professional team
- 1870, Red Stockings dissolve and many players go to Boston to form a team, though the franchise itself is not officially carried over
-1876, a new Cincinnati Red Stockings team becomes an original member of the National League
- 1880, the Red Stockings are expelled from the NL for serving beer and playing on Sundays
-1881, a third team of the same name becomes a charter member of the American Association
- 1890, the Red Stockings are welcomed back the National League. This is today's team.

This may all seem like semantics, but when I mentioned that it technically made the Reds not the oldest baseball franchise, some were upset. I was told the Reds essentially adopted the history of the past Red Stockings clubs, similar to how the old Browns records transferred to the Ravens, but... did they?

I see that the Reds claim the accomplishments of those clubs on their website, and I do indeed own an MLB-licensed t-shirt that reads "EST. 1869," but does anyone know if the wins/losses accrued by those original Stockings are counted in Reds franchise records? I would certainly imagine that the time the team(s) spent outside of the NL or in the AA would not 'count,' but are the seasons from 1876-1880 part of official Reds history? The Reds wiki page does state an 1881 establishment date.

I'm sure this discussion has been had on this board before, but I've been curious and I'm not sure of a quick way to check.