Rear derailleur has trouble switching gears on the way up, but works perfectly on the way down? I noticed an issue with my gears switching recently so I decided to try indexing them. I've gotten to the point where going from the smallest cog to the largest works perfectly, but in the other direction it doesn't want to move from gear 1 to 2. If I keep shifting up it shifts normally, only a gear behind, until it gets to gear 8 and the chain skips the 7th cog. Does this point to an issue with the derailleur itself?
This sounds like classic derailleur cable friction. The cable gets stuck inside the housing and stops or delays the derailleur moving to smaller sprockets. This is caused by dirt contaminating the housing, or perhaps corrosion of the cable.

The fix is to remove, clean, re-lube and re-install the cable, or replace the cable and housing altogether.