The old toilet is not like the ones shown in your link. I will have to look at the toilet again, its outside on the grass, to see if it looks normal. I assume it is a regular toilet. Waste out on the bottom that sits over a regular 4" drain pipe and flange with a wax ring. Normal. I will measure the pipe when I get back over there. Pretty sure its 4", but I will measure.
I will try to take a picture of the waste pipe in the floor and figure out how to post it. Hopefully. I have not removed the rubber cone. That is generally not something you happily run your hands over. But I will probably have to monkey with it. The drain pipe is plastic. But no flange was attached and then cut off. No.
The house is from the 1970s. It is one of those build in a factory houses. Where they build the two halves and truck them to the site and then lift them onto the basement foundation with a crane and then bolt the two halves together. And finish the very top of the roof ridge with shingles. So when it was built the bathrooms were finished except maybe the toilets and maybe sinks were not installed or hooked up. And then a plumber came in after the house was set on the foundation and did all the final water and drain pipe running in the basement. And maybe electrical connections too. The basement joists are covered on both sides with plywood. And the finish plumber cut out chunks of the plywood to run the pipes up to the bathrooms where they needed to be. The water pipes to the bathroom sinks and toilet are straight up through the floor. Not into the walls and then the pipe/valves out of the wall. A hole for the water pipe, and the toilet drain, straight up through the floor. So maybe it was easier and quicker to just drill a circle for the toilet drain pipe. Put the pipe up through the floor, put a rubber cone on it. Then screw a separate flange to the subfloor. Call it done. Put the toilet and wax ring on it. The bathroom sink did have the drain out of the wall. But it was a very big hole in the drywall. So I can imagine when the sink drain was hooked up, the plumber busted a huge hole in the drywall and used a long drill bit to drill down to the basement. Then the guy in the basement used a hole saw to make a big hole for the drain pipe that went up through the base plate of the wall and into the wall cavity.