I used to work on countertop induction cookers. There are several questions in your original post. I'll try to take them in order.
"My question is: do all induction cookers work the same way? Are they “binary” like this? Pulse wave modulation?"
Most work this way at low power settings. An induction cooker basically has electricity sloshing back and forth through a coil at about 20khz to create a field. As you can imagine if you only put a few watts through the coil the field is very weak (too weak to interact with the cookware that is maybe 1cm away separated by glass). Instead, to simulate low power settings most cookers modulate on and off. Do "all" work this way? I hesitate to answer yes to any question like that, but it's pretty safe to say that all consumer grade, affordable countertop induction cookers work that way.
If an induction cooker is working at, say, 70%, does that mean it’s just going to run at 100% for 7 seconds and then turn off for 3 seconds?
As you get to higher power settings the on-off modulation is generally replaced by varying the amount of current going through the coil. At 70% I'd expect a high quality induction cooker to be on 100% of the time but to vary the current. The burners I worked on functioned in this manner.
And if so, is there a difference in quality between induction cookers? What would make one better than another? Or is it just a matter of what cookware I use?
Induction cookers (like any electrical device) vary greatly in quality. Electrical components, coil construction, algorithms... They can all be changed depending on the specific cooker. Most cheap cookers come out of an area around Guangzhou, China. Search on Alibaba and you'll see lots of manufacturers that offer models differing only in cosmetic appearance. Higher end units are often produced in Europe or the Americas. That doesn't mean you can't get a good cooker from China, just that you have to be careful.
The cookware you use does matter too. Make sure you are using something that is strongly magnetic to get the most heating possible. As an aside, Panasonic made an induction cooker that works with all types of cookware (aluminum and copper too) by greatly increasing the khz going through the coild. Search for Panasonic Met-All if you are curious.
Or do some of them have better algorithms? Would it be better if it heated up and cooled down more rapidly, like 100% for 0.7 seconds and 0% for 0.3 seconds?
This gets a bit more technical. "Better" is such a tricky word. There are several ways to turn high voltages on and off quickly -- relays, solid state relays, triacs, FETs... Which you choose comes down to money, speed, durability... Sure, you could turn on and off faster as you suggest for more even temperatures. The problem is you tend to wear out the switching components (this is also why your electric oven cycles on and off). As an engineer you try to optimize for the expected life of the product when developing the switching algorithm.
Should I just buy a thicker pot?
Maybe? If your goal is to simmer a stew very evenly over a long period of time at a low temperature a thicker bottomed pot will help greatly. The greater mass will dampen the on-off behavior of the burner (a cast iron dutch oven can be great for this). If you are trying to sear a steak thinner bottom cookware is often the best. The thinner bottom allows the induction cooker to quickly add heat to the cookware once you add the steak. In other words for searing you often want responsive cookware and for simmering you want dampening cookware.
but if you already have induction with appropriate multi-layer pans you probably know that.
