there are a couple different types of chain:
1/8".
3/32"
9- and 10-speed chain. this **** is super narrow and we're not going to talk about it.

Now, cogs and chainrings that are made for 3/32" chain will work with a 1/8" chain just fine. but a 3/32" chain is not going to work on a cog or chainring that's made for 1/8". why? because you can't put a size small sweatshirt on a linebacker - follow?

lots of cogs come in either 3/32" or 1/8". it sounds like you're not sure which you got, and it sounds like you got 1/8". i don't know why this other bike shop's 1/8" chain didn't mesh on it, and i certainly don't know why there's a 3/16" chain in the equation - that's overkill.

there's also an issue of wear. i put a new chainring on my bike with an oldish chain, and it sounded horrible and every now and then the chain would delay before snapping into the chainring. felt the chainring for burrs, but that wasn't it - the chain was worn. "stretched," as they say, which just means that the pins get worn down so that the chain really is a little bit longer than it should be. when cogs, chainrings, and chains wear together, that's fine until you throw something unworn in to the mix - then their system is polluted and you need to start over. to avoid that, you want to put a new chain on when it gets all stretched out, so it doesn't wear your chainring and cog to the point where any other chain won't work well.

anyway: rocketrings will take a 3/32" chain, but they'll also work well with 1/8" chain - they're beefy. with your working 1/8" chain, you'll have a setup that should work fine.