Who said 200mA is the max current for an Arduino uno?

[PerryBebbington]

This wasn't written with your question in mind but I think it will help:

Common ground and why you need one

A common mistake for people new to electronics is to have 2 circuits fed from different power supplies with signals passing between the circuits, but with the grounds not connected. This leads to questions asking why the project doesn't work. You don't have to spend long reading the questions on this forum to see this is asked often. Here I attempt to explain the problem and why it matters to connect the grounds. Note that in all of the schematics that follow I have used a battery as a power sup…

@nitrosop

if current can flow in just 1 direction through a wire

Current can only flow one way through a wire. I'm not sure that's what you really meant though, see above tutorial.