The official spelling of the club/series name is something that has been ambiguous to me for a while, because I have seen it spelled all different ways in different sources. However, I have up until now almost exclusively used the spelling 'Baby-sitters Club' on this wiki because that is the spelling that appears in the books, so that is the canon spelling. I know that the logo makes it ambiguous, but I would consider the book text to be the best source in terms of what is canon.
However, the Netflix show appears to spell it 'Baby-Sitters Club' and that spelling is used on their official site/youtube channel. So, when referring to the show we should capitalize the 'S', but when referring to the books or anything within the books, the 's' should be lowercase. I will be making those changes to pages when I see them. Comments welcome.
Edit: Also, I've seen 'Baby-sitter's Club' (this is definitely wrong) and there is the whole issue of 'The Baby-sitters Club' vs just 'Baby-sitters Club'. Anyway, this issue drives me crazy.