While I was writing, artk posted that very wise post. Artk, I want you to be my guru.
OP, I think that your best approach depends upon your most important objective. There are lots of possibilities, all valid (or at least arguably valid, but that's not the point), and an approach that serves one can disserve another. And most are pretty big, too big to tackle at the same time anyway.
What's your primary objective here?
1) Avoid conflict
2) Resolve conflict
3) Explain your religious beliefs to your mother so she will understand
4) Defend your religious beliefs to your mother
5) Establish yourself as an autonomous adult
6) Win a power struggle
7) Get out of this endless loop
Can you see how some of these can conflict? It doesn't mean you can't work on all of them. Just maybe not at the same time. And the best approach for one just cannot be the best approach for all.
If you can let go of needing to win the point with her -- and I don't mean that all judgmental like I know it sounds; that is not easy, and it's a very legitimate feeling both for an adult child trying to get recognition as autonomous and for an adherent of a misunderstood and/or maligned religion -- then I think your path is smoother.
Besides, as someone else wisely pointed out upthread, at some point you are going to have to tell her that you have adopted this religion. Frankly, it's kind of not fair that you are letting her go on like this when you know that conversation is coming, but she doesn't. I'm not saying you have to have that conversation right now, but to be discussing Religion X without her knowing that is a bit less than an honest conversation. More important, you don't want to poison the whole subject before that important day.
Do resist contradicting her -- no, not because she was right to yell at you not to contradict her (of course she wasn't), but because there is no point. To her, it will just sound like it's about you calling her friend (or her) a liar or an idiot or both. If she brings up the puppy kicking, try to get out of it fast and non-confrontationally -- maybe something like, "Well, maybe she did back then, but I've looked into it, and it's not happening now" and/or "Mom, you know I'd never hang around puppy-kicking! You raised me better than that. Bean dip?" I mean, look at the puppy-kicking from her point of view: I believe you that Religion X does not kick puppies. But she doesn't; she thinks they do/did. She loves you and doesn't want you to get mixed up with puppy-kicking (and especially if what "puppy kicking" stands for is not just something distasteful, but something that is potentially dangerous to you.) You're an adult, but she still loves and cares for you and wants to try to protect you. For example, I know that many Christian parents are terribly upset when their children convert away, because they truly believe their children's immortal souls are going to spend eternity in hell. I don't believe that, but I don't blame them for feeling upset and for wanting to dissuade their children -- what kind of parents would they be if that didn't matter to them? It sounds like "puppy kicking" stands for something very different from that, but I imagine a similar emotional response is what's going on here. And if it WERE true about the "puppy kicking," she WOULD be doing the right thing by trying to alert you. (And not just because she's your parent -- the same would be true if you were the one trying to alert her to something.) So do consider her motives.
If you want to avoid conflict, POLITELY refuse to talk about it. I think that even "I'm not discussing this with you, because every time we do, you refuse to listen to me" is a little bit unnecessarily confrontational, because it talks about and criticizes HER behavior. Yes, accurately! And not meanly. But that's not the point. When we do that, the other person stops listening, because they feel criticized and defensive. Rather, try to talk in terms of your OWN feelings, not her behavior that is causing them -- what some therapists call "I" messages. (Like, in another context, a spouse will do better with "I would like more help with the dishes" than "You never help with the dishes.") Talk about how you feel and what you need and want, not about what they are doing wrong.
So in your case, you might tweak that just a bit to something like, "Mom, we've gone around and around on this so many times, and I feel really frustrated because we never get anywhere. I know you're just worried because you care about me, and I want you to know I hear and understand your concerns. I would really appreciate it if we could just avoid this subject for a while."
Whatever approach you choose, engagement or avoidance, don't be rude. Be gentle. The fact that you are 100% right doesn't make it okay to be rude to your mom, even if she is overstepping. You may reach a point when that's not possible, but so far it doesn't sound like it. Resist the temptation to tell yourself that it's okay to be rude and aggressive just because she started it. The payoff is that being polite and respectful will serve your other objectives -- being seen as an autonomous adult and as a credible source of valid information about Religion X -- much better than appearing, in her eyes, like a fractious teenager.
Good luck. I'm sure that this is really hard. No matter how carefully you do it, it's a big challenge -- so give yourself a good start.