Bingo. Just as if you wanted a better vertical jump, you practice plyometric vertical jumps. Sure, maybe working on bigger legs with leg presses will help a little, but it is a roundabout way to approach your goals.Yeah it is a progression thing. Whether it is 2 months of boxing or 2 months of karate you will be more well off then if you were to be without the 2 months of boxing or 2 months of karate.
I don't think the weight lifting will help you at all in a fight though. A punch or a kick isn't all that about arm or leg muscle strength. It's more about the physics of how you distribute the force of your feet moving off the ground, thougout your body, and finally into your fist. The strength would really only come into play if you are grappling and even then technique is more important then actual strength. Hip rotation is also very important.
My point is it does help, but the training in the martial arts whether it be boxing or a more asianic style, that would be much more important.
If you go to a strength trainer and ask about strength training, he or she is going to ask about your primary goals: functional strength, "explosive" strength, flexibility, stamina or size/hypertrophy, and the routine is going to be a little different for each. Most bodybuilders I've talked to avoid cardio like the plague . . . I don't think any sports fighter would do the same.