The first option seems fine but be aware that you will increase the water flow to the lower roof gutter a bit and the run in that gutter to the downspout looks long so just be sure it's properly designed. It doesn't look as if the original designer thought much about the gutter system so check it out.

If it snows much were you live the gutters should be set low enough that a sheet of ice sliding down the roof would not contact the outer lip of the gutter.

I just put on some gutters and have a somewhat similar situation with one roof draining on to another. I was looking at how other homes configured theirs. It was about 1/2 and 1/2. The only issue is that on homes that had the 1st config. that were older (most homes have asphalt roofing here in Mn) there appeared to be more wear and tear on the shingles due to the rain running down the lower roof to reach the lower gutter. I went with option #2. But if you don't get much rain it may not be an issue.

I would drip it straight down from one gutter into the other.

Be sure to use some flashing to prevent splashing were the downspout empties into the gutter.

The first option is sometimes preferable, I have seen a tee added with a horizontal piece with several holes in it to spread out the flow of water.

Draining into the second gutter will probably look better and avoid the wear to that spot of the roof, but there is the risk of overflow.

Keep in mind, if you do the second option and find it overflows, it's not too hard or expensive to change.

"The first option is sometimes preferable,..."

Dropping a downspout onto a shingle roof is rarely preferred.

"I have seen a tee added with a horizontal piece with several holes in it to spread out the flow of water."

This will quickly clog with ANY debris that makes it into the gutters.
Unless you want to climb up to clean out the filter you have created, stay away from this.

Once you have collected water from a gutter into a downspout, try to keep it confined.

Didn't see they were shingles. I've never seen the arrangement I've described clogged, you leave both ends open.

"...you leave both ends open."

Then with a large rainfall the water ss going to come out there (even more so after the smaller holes clog up with debris).

Not a very effective way to control the runoff.

Water running sideways on shingled roofs is a common cause of leaking (look at how high valleys are woven or why flashed valleys are formed to try and limit any sideways runoff).

Our previous house had this setup. Go option one and when the downspout hits the lower roof continue the downspout till you get to the next gutter. It will lay on the roof. Ours ended with an elbow in the direction of flow just above the second gutter. Water never ran on the roof from the downspout.

I would do like what you have in the second picture. I would send the water into the gutter flowing in the direction the gutter would flow whenever possible.

Had water from upper gutter dropping directly onto the roof of addition( option 1) Had water leak into the addition due to this. I'd recommend keeping the water in a downspout that routes to the lower gutter.