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.
Honestly gutters are something I pay for. Most good companies have machines that bend and shape them on site so you can make them continuous. That prevents leaks and ingress.
They also will have it done in a day, doing it my self would take much longer.
This chunk of roof is so short that there’d be no difference between hiring a company for continuous gutters and DIYing a segment of gutters.
I see they’re trying to follow the line of the eaves, but the more accepted way of doing this would be a bottom outlet on the upper gutter, an elbow to attach the downspout section, then empty the downspout into the top of the lower gutter with another elbow. I’m not sure how the arrangement you show would avoid clogging and leaking. Probably not well.
