What is your relationship to your mom? The way you describe it, it sounds like she is putting you down which is sad and not ok. Shouldn’t she help you learn if she feels you don’t know? What is the point of giving you one day to prove yourself? But maybe there is context I am missing.
As others have said, you develop your ability to cook through repeated use, trial and error, and ideally knowledge passed down from your family. If you want to become someone who can cook, it won’t happen after one try or ten. Just keep trying out recipes that sound good to you, and treat each one as a learning experience. Overtime you will build up recipes that you have memorized and can riff on confidently.
You must like to cook, since you want to prove you can. Whatever you make, take your time, trust your gut, and do it to feed yourself something you like. Cooking is a personal thing, just because not everyone likes what you make doesn’t mean you can’t do it.
I think you are doing great, good on you for thinking about what you eat and doing your best to do it thoughtfully. I think you should feel comfortable calling yourself vegetarian because you are doing your best to eat as little meat as possible. People who ask questions are probably mostly curious how it works and sharing your experience might help them make similar choices. In my experience the only people policing these labels are too online or openly antagonistic to the entire premise I don’t pay either much mind.
Personally I don’t like eggs much, but I think getting then from a local farm you have a relationship with is the best possible scenario. I do try to avoid dairy as much as possible. You have to remember, in order to get animal milk you need to take it from the baby animals. At scale this involves a pretty horrible love for everyone involved. At small scale maybe it’s ok. If the mom is overproducing for example, and both the mom and child have a good life. But that doesn’t exist in the grocery store or even at most farms. So I just avoid it. It’s hard at first until you find good replacements but many cultures in the world got by fine without it for centuries so there’s plenty of great recipes to learn. I’m not interested in trying to get everyone off it, but most people I talk to have never given it a thought. They think cows just make milk no matter what and need you to milk them so we might as well. I think if you know the true cost of something you might at least have more respect for it.
But as far as things like bone broth, gelatin etc. I basically agree, these are waste products. I might not like the industry it’s coming from, but for God’s sake if you are going to kill an animal at least use every last bit of it.