Robert, I have a brother printer and I can either set it up to distinguish between a fax and a phone call or to just manually answer a fax when I know one is due to come in. I opt for the latter as I get maybe 6 faxes a year
I don't have a modular phone splitter (not sure what that is, lol) I just connect it with the phone jack cord that comes with the printer.

What I like about Brother printers is that I can get the very expensive replacement ink on Ebay. It's generic ink but I can't tell the difference between that and name brand ink. I get a case of 4 blacks and two of each color for under $20.