Just a heads up, not saying you don't realize this and maybe it's obvious, but it wasn't to my wife and a friend I have. The problem has nothing to do with the floor drain, that's just the lowest spot, so when it backs up, that's where the water comes first. I came home one day and my wife had taken the cover off the drain and had been rooting around in there and said she couldn't find the clog. I had a friend callme saying he did the same thing and if I had any advice finding the clog in the floor drain. So, just throwing it out there in case you don't know anything about plumbing.
Washing machines create a huge outflow of water all at once, compared to almost anything else you will do, even shower. You have a partially clogged pipe somwhere in the drain system. More than likely, somewhere in your yard. I would definitely call the pros and have them go through the clean out.