It definitely looks like an indexing problem, where the derailleur doesn't sit quite right.
Assuming your shifter has the same number of "positions" as there are cogs in the cassette, then it should be fixable.
Good work hanging the bike up - that helps. Start by slow pedalling with your left hand on the right pedal, and use your right hand to click the shifter.
You should be able to get the chain to click up a cog with every shifter click, and down again with the other lever.
If the chain jumps down two, or clatters and tries to come down early, then unscrew the barrel adjuster by a half-turn (effectively . If the chain tries to climb up the cassette too soon, then tighten the barrel connector a bit into the rear derailleur. Work up and down the cassette while slow-hand-pedalling and find a median position where everything works.
If there is no position where everything works, then another common cause is a subtle bend in the derailleur hanger or the derailleur itself. Has the bike ever been dropped on its side?