Looking to put gutters here. 
This is how others next door are done, as well as how the ones I'm replacing were done previously. Basically a downspout connected between two gutter's via a hole cut in them.
Is this actually the correct way to do this, or is there a better way that doesn't seem so janky and leak prone.
I would drop a downspout from the second story then do another from the one-story section. Hard to tell from the pics but if I'm seeing it right that second story run appears to be over 20 feet. You kind want to drop the water at least every 30 feet max.
It's all pretty low. Building on right is a shed and theres a 3 ft section that's part of the carport. That drained down a downspout to the lower end of the shed which drains to the ground.
Downspouts on the upper short carport gutter would have to drain into the driveway, so not ideal.