I've been very active in sports all my life, including weight training, and I've never noticed any detriment to my playing. In fact, when I am the most diligent in my activities, I find that I have more power and stamina than ever. The only activities I won't engage in are chopping wood and bowling. I had a week-long injury from bowling once...my right hand only hurt when I played a 1st inversion Eb chord from g to g. Weird.
If you train with weights *properly* by doing controlled, deliberate reps and not alot of jarring and yanking, then you should be fine.
You mentioned putting strain on the fingers...the act of playing the piano puts alot of strain on the limbs as it is. If one practices for few hours, more than likely their hands and arms are a little fatigued after the practice. This is because you have "worked out" your fingers and arms. Everytime a finger moves, a muscle is being worked.Practicing is more than just learning notes and the proper mechanism for playing the piano...it is just as much strength training for our bodies to keep our hands in shape.