I mean, there had already been a The Bill Cosby Show (which was also telecast on NBC) in 1969-71, and a The New Bill Cosby Show during the 1972-73 season (but on CBS).

Evidence of this was all of the "Ratings Roundups" on issues of Broadcasting magazine until midway through The Cosby Show's first season (specifically the January 21, 1985 issue (which recorded the ratings for its January 10 original airing) (https://worldradiohistory.com/hd2/IDX-Business/Magazines/Archive-BC-IDX/85-OCR/BC-1985-01-21-OCR-Page-0089.pdf#search=%22cosby%20show%22); and this series pre-premiere promo on YouTube (https://www.youtube.com/watch?v=DcgkInHPZNc&t=52s) (skip to 0:52); and a couple of (https://worldradiohistory.com/hd2/IDX-Business/Magazines/Archive-BC-IDX/90-OCR/BC-1990-02-12-OCR-Page-0008.pdf) Broadcasting issues (https://worldradiohistory.com/hd2/IDX-Business/Magazines/Archive-BC-IDX/87-OCR/BC-1987-10-05-OCR-Page-0058.pdf) from Fall 1987. This Spring 1990 issue (https://worldradiohistory.com/hd2/IDX-Business/Magazines/Archive-BC-IDX/90-OCR/BC-1990-04-23-OCR-Page-0016.pdf) of Broadcasting (from April 21) was the publication's last reference to TCS as "The Bill Cosby Show" (after two (https://worldradiohistory.com/hd2/IDX-Business/Magazines/Archive-BC-IDX/90-OCR/BC-1990-02-05-OCR-Page-0014.pdf) issues (https://worldradiohistory.com/hd2/IDX-Business/Magazines/Archive-BC-IDX/90-OCR/BC-1990-02-12-OCR-Page-0008.pdf) from the previous February did the same).