Like most other parts, Shimano rear derailers are labeled "8-speed", "9-speed", etc, but really they haven't changed much at all in the last 20 years or so, except for some of the newest MTB RDs.

The 600 RD will handle a 9-speed cassette just fine. I would expect either RD to work equally well. The only issue you might run into is front shifting. The narrower 9S chain can "skate" between the two rings when you try to shift into the little ring, but this should never be an issue as long as you don't shift the front when standing up (which you should never do anyway.)