I have an older Shimano Ultegra Shifter that broke, and I am wondering how I can know what shifters are compatible with the gear set as a replacement? My broken shifter is on the right side and is a Shimano Ultegra ST-6510 9-speed. Are current generation Shimano 9-speeds backwards compatible with my older gear set? I've looked at this Sora shifter as a possible replacement. I am not so concerned about weight or performance and am only hoping to get a couple more years out of my older bike.
Any Shimano compatible 9 speed road shifter like the Sora you linked will work. While it will be visually different it will perform the same function. The last of these models were produced in 2007. While the Sora shifters you are looking at may be lower on the hierarchy list, they are likely equal to or better than your nearly the 20 year old technology you currently have. As new improvements are introduced Shimano eliminates the oldest technology and everything moves down a tier. Latest version of Ultegra is the previous version of Dura-Ace and it all moves down the line. My guess is you will be satisfied with a new Sora shifter.