All Shimano indexed shifters and rear deraillers are compatible with each other, except for pre-9-speed Dura Ace. There are some variations in front derailler indexing, which matter for STI and Rapidfire shifters but not for down-tube and bar-end (which both have friction shifting for the front der).
The Shimano instructions for my 9-speed bar-end shifters said "compatible only with Dura Ace rear derailler" which is either a holdover from 8-speed Dura-Ace bar-end shifters and Shimano forgot to remove it when they printed instructions for 9-speed bar-end shifters, or it's an intentional crock to try and get people to buy superfluous updates to ensure compatibility where it wouldn't have been a problem anyway.
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"c" is not a unit that measures tire width