You wanna know how "Cheers" is a "Taxi" spin-off? I'll tell ya. You see, between the year 1978 and 1982 (when "Taxi" was on ABC) Ted Danson, Rhea Perlman, and George Wendt appeared on different episodes of "Taxi," each in their "Cheers" characters. The only difference between Sam on "Cheers" and Sam on "Taxi" is that Sam was a barber on "Taxi," not a bartender like he was on "Cheers." If you see an old Nick at Nite promo of "Taxi" from around 1996-ish that'll explain it to you much better.

http://timstvshowcase.com/cheers.gif

http://timstvshowcase.com/cheers2.jpg

Wait. When George Wendt was on Taxi he played an exterminator. When Rhea was on she played Danny Devito's Girlfriend, I think her name was zena or something. Danny and Rhea are husband and wife in real life just in case you didn't know. Not sure about Ted though. I don't see the connection.

Cheers is not a spinoff of Taxi. The principal director on Taxi was James Burrows.
James Burrows, along with Glen and Les Charles, created Cheers. He was also a principal director on Cheers.
This was simply a case of James Burrows casting and directing Ted Danson, Rhea Pherlman and George Wendt on Taxi--liking their work and then casting them on his own show, Cheers.
The Nick at Nite promo that you're referring to was probably one of their comical gags.