This is why people shouldn't talk to The Aprons about anything to do with toilets and instead should come here. That marketing piece at HD on "How to Repair A Toilet" is useless and hideous.
Okay. First, I can't tell from the photos for sure, but it definitely seems to me that the water level in your tank is right to the top of the overflow riser. (I'm getting this from your photos 1805 and 1806.) That's your problem.
[The overflow riser is that white plastic tube that is sticking up with the black (in your case brown) hose running into it. It runs down next to the flapper, and any water that goes into it bypasses the flapper and goes down the flush hole into the bowl below. It serves two purposes: First, water to refill the bowl runs from the fill valve through the black hose and down the overflow riser into the bowl. During the period that the tank is refilling, and only then, water will run through the black hose, through the overflow riser, and into the bowl to refill the bowl. Second, if the tank is overfilled (i.e. the fill valve fails to shut off), instead of pouring out the top of the tank, the water ("overflow") will run down that tube and into the bowl.]
If the tank is set to refill right to the edge of that riser, you're sometimes going to get a smidge of overflow, which will go down that riser and maybe cause that kind of fill valve (ballcock) to put a teensy bit more water slowly into the toilet, and it becomes a vicious cycle. If you set the ballcock so that the thing refills to about 1/2" below the top of that overflow riser, which is where most toilet manufacturers want you to set it, I think your problem will be solved. The way to do that is to bend the metal rod with the ball on it DOWN a little bit (hold one part of it in your left hand so you don't snap it off the valve, and push down a bit at the end). You can also adjust the screw on the valve, but most people just bend the rod.
One advantage of the Korky Quietfill valve (also called the 528) is that when it shuts off, it shuts off completely and doesn't allow water to run again until the tank level falls about 1/4". That way, little bits of water don't slowly run to refill the tank when there is just a little leak or little overflow, as with the traditional ballcock that you have. But there's no reason to mess with the fill valve unless it has suddenly failed to work properly, and you have given us no indication that it has. (On the other hand, the brown stain in the tank is about an inch below what now appears to be the waterline, so perhaps the valve is not fully shutting-off anymore...) Another advantage of the Quietfill is that once you install it, it is easy to change the water level in the tank.
Also, the next time you replace the flapper, chuck that Fluidmaster flapper and use a Korky flapper instead. I think you'll find it easier to to replace and it should be more reliable.
Let us know if we have found your problem, and don't be afraid to come back for more help.
If you do decide to replace the fill valve with a Korky Quietfill, there are helpful videos at korky.com on how to do it, or come back here and we'll give you a hand. It's going to take you ten minutes to do, most of which time will be spent trying to disconnect the old fill valve.