Wait, how does a plate plugging the sink drain cause the dishwasher to fill up with water from the sink?
There's several different scenarios, depending on how your dishwasher (DW) drain returns to the sink's drainage system (and how many sinks you have). There are three ways (at least?) to plumb the DW drain. One is like yours, directly into the drain system. The second goes first into a garbage disposer (GD), which leads into the drain system. Of the two, into the GD is preferred, only because the GD can chew up any large chunks of food before introducing that waste into the sewer system. But neither of those routes is affected by how many dishes are in the sink, or if the sink is stopped up.
The third route is through an air gap fitting over your sink. That MO is a two-fer. The DW waste routes through that fitting such that if the drain system ever got backed up, the DW waste water would empty into the sink, rather than back-flowing back into the DW. The best way to plumb the air gap is to have the gap over one sink, and the exit hose from the air gap plumbed to the drain (or GD) of the
other sink. So say the GD gets clogged up, then the DW waste would back up but exit through the air gap (instead of back flowing to the DW) and drain into the other sink. If that sink's drain is free, then the DW waste goes through to the sewer. But if both sinks are backed up, then the DW waste will collect in the sink, hopefully not overflowing it before noticed. So a correctly routed air-gap setup is the safest, because there are several places the DW waste can end up before it gets back to the DW or onto your floor. Whew, OK...
One way dishes in the sink matter is if the air gap gets used, and the DW waste gets trapped in the sink due to dirty dishes or a closed strainer or otherwise blocked sink drain.
As mentioned, you're supposed to run the GD right before you run the DW, clear the path between the air gap and sink, and be sure all the drains are open and ideally no dishes in the sink. But that's if you have an air gap. You don't, and your DW drain doesn't run to a GD, so none of the above applies to you.
But if you were inspired, you could redo that. Your "worst case" is DW food waste routed directly to the drain pipe, which could get backed up, and then you'll have a real mess. It might backflow into the sink, if that sinks drain is clear, or it might backflow to the DW. This is the other reason a sink full of dishes blocking the drain matters: but only if the drain pipe is clogged. I don't know that the loop under the counter is going to matter much in that scenario.
The loop under the counter is really only for keeping the sink water from getting to the DW, which it does, unless the level of water in the sink is higher than the loop. Which is why an air gap over the sink is better for that, too, because the air gap is never going to be lower than the water in the sink.
So... the best way to plumb the DW drain line:
DW --> air gap --> GD, with the air gap fitting over the non-GD sink if you have two sinks.
If you don't have a GD, then it's:
DW --> air gap --> drain pipe of sink #2, with the air gap fitting over sink #1, if you have two sinks.
