For many people, "full" feeling is usually triggered by fat rather than by the mechanical stuffing of the stomach. Red meats tend to be higher in fats than poultry and fish.
adults are advised to consume on average no more than 70g of red meat per day.
Poultry and fish are not red meat. Pork, sheep and beef are the most common red meats that you will encounter in a modern western diet. If you are concerned about getting enough protein, one could add tofu or tempeh to one's diet, although tofu can be high in fat.
Chicken and Fish! Especially fish. Fish contain great deal of "good" fat and lots of protein. When you get hungry snack on nuts, protein shake, or protein bar. Lots of veggie. If you need rice, eat brown rice. I know it taste like plastic, but the best way is to put some scrushed seaweed seasoning on top (not too much). It'll make it taste better.
There have been findings that suggest we cut back on red and processed meat consumption as they are being linked to reduction of life span and risks to overall health. Though correlation does not imply causation, I don't think it hurts to cut back esp processed meat. As a fellow die-hard carnivore, I have been substituting lean ground turkey (white meat) in place of ground beef in chilli and burger patties.
Seafood and poultry are not red meat so you are not intaking as much red meat as you think.
I also agree with what was said about that article not being serious or backed-up by science, but I just want to add that if you are feeling hungry just from cutting down on the meat, you might also want to watch the other aspects of your diet. Energy should come from carbs and/or fat, make sure you are getting enough of that as well. You don't want to be burning protein for energy.