Six speed chain on an eight speed bike
The skipping will always be because of the chain touching on the other cogs, or trying to slip back down from the pull of the rear derailleur. The chain is too wide and, as you already know, is doing one or the other. It might not be evident from looking at the cogs while riding as the chain is fed from underneath when you pedal, so the clicking from the chain moving may only be visible when the bike is upside down.

If you have a reasonable quality shifters/cables/mech then you should be able to just about get away with it with some really fine adjustment.

The difference between 6, 7, 8 and 9 speed is actually the spacing between the cogs, and not the width of the cogs themselves. Hence the need for thinner chains.

Edit: I haven't got comment everywhere ability yet, but as suggested below, you should make sure your chain is the correct length. The recognised method for this is to wrap the chain around the largest cog at the front, the largest cog at the back, pull tight and then add two links (or 3 if it ended up with two inner links).

This generally ends up being shorter than what you had originally, but is correct.