If it is overflowing at the basement floor drain, then the blockage has to be downstream of that point, so that would be a good point to try snaking from. Since it is not happening with regular sink, shower, dishwasher, etc flow you only have a partial blockage that will let 5 gallons or less though OK, but not the 10+ gallons all at once from the dishwasher. I would guess a bathtub emptying causes the same bubbling up.
If snaking does not work from the floor drain, then try a point further downstream. Do you have a downstairs toilet, garage floor drain, etc downflow from that floor drain. Frequently semi-floating solids and grease build up at the wye's to that sort of drain and block the intersection, especially if it is not frequently used, as with a basement guest bathroom (or stubbed out and unused basement toilet or tub connection) or garage floor drain. Try snaking and flushing from those points if the floor drain snaking does not work.
When snaking be sure to have water running somewhere upstream of where you are snaking, so what you scrape off goes down the pipe immediately - else you can turn a partial clob into a full blockage. It helps to have someone else watching any basement drains when doing this, so they can run and shut off the water if it starts backing up on you. Also, if you have a crawl space, there may be cleanout plugs there you can snake / flush from.
Typically, there will be a vertical cleanout pipe very close outside the house as well (typically about 3" or 4" cast iron or black ABS plastic; may have a square-head xxxx -in plug or more likely a rubber cap held on with a stainless steel scre clamp). Remove the cover, and take a strong flashlight and shine it down there at a time the drain is backing up to see if it is backing up in that riser - if so, then your blockage is between the house and the street, not in or under the house (unless there are two). If the blockage is in or under the house, you will see a flow dropping into the riser where the sewer comes out from the house, and flowing freely down the riser into the lower (about 6 feet deep, typically) sewer run to the street. At least initially when you start running water full force, that flow will be less than is going into the pipe because it is backing up in the pipes under the house.
Having been down this road myself a couple of times, I have to say that snaking itself yourself may temporarily alleviate the problem, but only the full-diameter cutting edge of the power router used by a sewer and drain contractor really returns the pipe to full diameter, while also removing the grease and toilet tissue buildup in the pipe, which generally needs to be routed out every 5-10 years anyway.