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.

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.