For a while now, we’ve listed that Barkley debuted in season 9 (1977-78) of Sesame Street. Danny added it back in 2005 and it’s been too long to tell where that originally came from (the old “Sesame Street Encyclopedia” website maybe?), but I’ve seen several readings discussing Barkley that say he was created in 1978 for season 10 specifically. Here’s a source rundown:
Imagination Illustrated (pg. 114) includes a Jim Henson journal entry, dated January/February 1978, noting the development period of Barkley the Dog; apparently indicating when he was first created.
Sesame Street Unpaved (pg. 132) identifies 1978 as Barkley’s debut year (although it’s ambiguously worded as when he “joined the cast full-time”)
A news article promoting Sesame Street’s 10th season notes Barkley as "the new Muppet."
Toby Towson’s own account says he was enlisted to “model for the building of a new Sesame Street character which would debut for the tenth anniversary season [...]” originally conceived as an acrobatic ape.
As an incidental note for the latter, Toby's page says that he played the role of Barkley for only several months. Michael had mentioned on Talk:Elmo that Brian Muehl joined Sesame "during the second half" of season 10 in an e-mail correspondence. So, to correlate with this info, the performer date ranges should be changed to "Towson (1978), Muehl (1979-1984)".
Additionally, there's also this note on the season 10 page (sourced from the Fraggle Rock Season 3 DVD, which I also checked), mentioning that Henson and Frith were assigned to brainstorm ideas for new Muppets for the 10th season; with Frith first creating what became Marjory the Trash Heap.
Those are all irrefutable sources. I'm guessing Danny saw 1978 as his debut year and picked season 9. It's also notable that the first episode of season 10 features a story where the Street residents name this new dog friend they've made.
Scans from various script highlights for season 11 and 12 have recently made their way online. The first page for each week of highlights for season 11 has a special article about the show/season in general. The front page article for issue six (week of December 31, 1979 - January 4, 1980 - shows 1341-1345) mentions that Barkley joined the cast on a full-time basis in 1978 after making a few guest appearances in prior seasons, though it doesn't say what seasons (I'm guessing season 9 would be the earliest, but it doesn't say for sure).
So back to this - Episode 1177, a Season 9 episode, features Woof-Woof, which as we know was established as Barkley's early name in the Season 10 premiere. Whether or not this is meant to be the same Woof-Woof is up in the air until we can actually see the episode in question, I guess, but it's definitely a possibility.