Is it safe if you've put meat to thaw in the refrigerator and then put it in cold water if its still partially frozen?

Yes, that's pretty much how everybody defrosts a frozen turkey, which I will be doing tomorrow morning. It has been in the fridge for 5 days and it still feels partly frozen. It's going in the sink in the morning (cold water):)

OK thanks for the answer ;-), also can you leave it in the package as is, or should u put the package in a ziploc bag and then put in water?

I leave it in the package, if it's a turkey.
If it's red meat, you might want to put it in a zip lock. I guessing here because I rinse poultry, but not pork or red meat. I think it might lose flavor.:)

Hmmm, right.

OK, one more question, what if water gets into the package while defrosting it in cold water

I don't think it should matter, it might look different, but the taste would be the same. There are people who wash red meat.