the original design is by rolins. you can find sketches of bill HERE
the way i see it is like this:
because my head canon is that the batter becomes bad batter when he’s stripped from his puppeteer (thats my own personal inference in the reason why we see bad batter when we choose the judge’s ending) bill cipher sought out a puppeteer in cecil to keep the batter from becoming a loose canon.
the batter views they’re relationship professional (puppet to puppeteer) and although he seems pissed by cecil’s constant chit chats he’s truly amused by the other and is quite fond of him (just wont admit it). cecil on the other hand fucking adores batter and looks up to him like he were an older brother. he has a big infatuation for batter but in a respectable way, kind of like a child and his or her idol.