The problem is that when vegetarian diet includes diary products (not vegan), they are still produced by cattle or chicken. Which you evocatively described as bad for climate.
Much of your reasoning is weak because you mix completely different regions of the world and economies. It would make more sense if you concentrated on one area of the world. If you live in Belgium or Britain you normally buy beef produced in Europe. So your choices of eating meat don't affect conversion of land to raise meat cattle in other continents. You affect tropical biodiversity when you buy palm oil or coffee.
If one lives in Europe, then carbon footprint of diet and associated land use has little to do with the high-level ecological rules, such as that producing meat or diary takes more energy than grain. The system is skewed by regional and agricultural policy, with subsidies to produce more food than needed and artificially maintain low-quality agricultural land. Here you can help climate locally by lobbying to change agricultural politics, not paying farmers to produce more food than needed and let the low-quality agricultural land to return to wildlife.
And sorry, my low opinion about vegans comes from that I recently discussed with several preaching vegans, and they just could not understand that e.g. meadows in Europe are usually located on hills, mountains or near rivers, because this land could not be plowed for grain production. Or that biodiversity goes extinct primarily from habitat degradation not climate change.