First: Was the "installer" a licensed plumber? Or something else?
You don't need to be a plumber to install a toilet, but handy-hacks seem to screw up toilet installations more often than even DIY-ers. Reach and Jim explain above how that tank should be seated, and there's a tutorial on the Drake at the top of this section of the forum. What you describe sounds like every other tank I have seen where the person didn't know how to do it right and left it before it was sufficiently-tightened-down. In particular, that tank has to come straight down vertically, which sometimes means loosening, restraightening, and retightening if it's coming down at an angle. And one has to pay attention to make sure it comes down enough but not too much. The porcelain shouldn't touch, but it should be close. It involves some checking then tightening then checking then tightening. And when it's done the tank will be rock solid. I have installed several Toto Drakes, and with a little paying of attention and following instructions, the tank is solid.
For Toto to have shipped a defective tank isn't impossible, but it is extremely, extraordinarily unlikely. They have the best quality control and consistency in the business. That's why your supply house was skeptical. They deserve lots of credit for agreeing to swap tanks for free.
I'm voting for installer error -- and error that likely could be easily-corrected if the guy -- like all incompetents -- didn't keep insisting that he did it right.
Here's an acid test for whose fault it is: Did the "installer" put the thing in and right away say, "Oh my gosh! This tank is defective! I can't get it stable and it's leaking!! I will take it right away back to the supply house!" Or did he say, "There you go," take your check, and only start making excuses about the tank being defective when you called him to describe your problem? I'm guessing the latter. That he would have left your house with the tank flapping against the wall behind the toilet is an indication that he didn't do it right. I would get a different installer, or supervisor from the same company, or install the tank yourself. If the base is done correctly (is it?), then the tank part shouldn't be too physically-demanding.
Let me ask a simple question: when you look in the tank, did the "installer" put just the rubber washers under the bolt head? Or did he put the metal washers AND the rubber washers in the tank? That's a common handyman mistake, and it leads to leaks. If you don't know, shoot us a picture.
