Your picture shows pretty clearly that the narrow (3/32") chain has a severe problem fitting on the chainring - and looks just like what I'd expect if the chain ring is designed for 1/8" (i.e. wider) chain. The rollers of the chain aren't going all the way down into the space between the teeth of the chain ring. Therefore the chain is sitting above the proper place on the ring and as a result looks like the spacing of the links is too small. But if the chain were sitting lower on the chain ring the spacing would be just fine.

You need to get a chain ring made for 3/32" wide chain.