The various business definitions can be found using the dictionary website. Bussiness isn’t a real word, but busyness is… and that is the state of being busy. So website resources like Grammerly would point out the words as correct or as incorrect and usually offer correct spellings, or at least raise issues for consideration. It’s then up to the writer to decide. It’s not a good idea to add a bad way of spelling to any personal dictionary, so use dictionary websites and Google search results to find out about spelling and context examples, too. UK and US spellings are a matter of choice for some common words, so the decisions are not as easy as we first imagine.
To be generous to your sources, I am quite sure that “bussiness” does not exist in English. Why? The suffix” -iness” means “full of” rherefore we are stuck with the meaning, “full of buses”? I don’t think so. Sometimes clever words, like “truthiness” are coined and they later become useful but”bussiness” doesn’t qualify in my view. The word, “business”, on the other hand, is common and it’s meaning is clear. My advisce is to do yourself a favor: forget the non-word, “bussiness” and use the good old English word, “business”.