I think, according to that good video you linked, the holding in your lap etc is only for the first 3-4 days. After that, you can closely supervise indoor puppy time. Mistakes will happen. Catching them in the act simply involves some gentle repeated "No, no no, outside." Carry them outside even when you know the pee or poo is going to land on you. Once they are outside, use the "pee" command. I used, "Do your thing" lol.

I have also had to clean up mistakes of course. Instead of throwing the mess into the garbage, I picked up the puppy, again with the gentle no no (GENTLE) and took the mess to where they are supposed to go. Cleaning up the mess so the scent isn't there is essential but not hard. A spray bottle with half and half water and white vinegar with a teaspoon of dish soap works well. Enzymatic cleaners specifically to clean up pee and poo can work well but they are a little pricey. A small carpet cleaner is helpful. Get a rental with darker colored carpet if it has carpeting!!!

The most often used rule of thumb for crate training is age of pup in months is how many hours they can stay in the crate before taking them outside. A 2 month old pup can be in crate for 2 hours. A 4 month old pup is 4 hours etc.