Look, I know you're probably right on your central point: that you're mature enough to handle the violence from a game like MW2 without it having a negative effect on you emotionally or psychologically.
You can take pride in that fact. Yes, you're right. Yes, you have the maturity.
AND- this is absolutely, positively, NOT going to cause you to win this argument with your mother.
One thing most adults learn eventually, and even 13-year-olds with 134 IQs don't tend to know (sorry), is this: some fights aren't worth the cost of "winning".
I don't know when your father died, but what it sounds like is this: your mother wants to protect you. She loves you, and this is how she's expressing it. You're trying to win a "logical" argument, and you won't, because she's not having a "logical" argument, she's having an emotional one. And your attempts to win the "logical" argument may even feel like rejections of her desire to love and protect you.
In two years, I can all but guarantee you you will no longer care about Modern Warfare 2. But you may still regret it if you allowed this argument to raise barriers between the two of you.
My honest suggestion is this: as much as it hurts, as much as it feels "unfair", try to respect your mom's decision. Let her know that you understand that she's trying to protect you, but let her know, respectfully, that you're frustrated at not being able to play a game your peers do. But that you will respect and abide by her decision. And ask her for some guidelines as to what she finds offensive on the grounds that you don't want to risk such conflicts with her in the future; apparently, it isn't simply a matter of an "M" rating to her. Be reasonable, don't attack her decision, and you may be able to come to an accord on future gaming.