Is water overflowing the gutter at the edge of the roof, and crashing down through the air? If that's the case, then the gutter is undersized or there aren't enough downspouts. If water is not over-spilling the gutter, then the problem is not with the gutter or downspout. It is with grading at the grade level, or with drainage.

Suppose that by some form of magic, any water that hits the roof is vacuumed away, and we can remove all that water from the equation. For the water that hits the ground, how does it escape this area? In one of your darker pictures, it looks like there might be a drain in the walk outside of the house door. Is that the sole method of escape for water? Or is there another method such as slope, which water can use to escape?

In the last picture, are you asking if there should be a downspout added? The answer would largely depend on what is happening at the ground level ... where the downspout empties.

First thank you Yardvaark very much for the response.

Answer to first paragraph questions:

The water is indeed overflowing at the edge of the roof and crashing down through the air (in the second picture above I've circled where this happens). This is the central problem (as far as I can tell).

Second paragraph:

Apart from the water that overflows off the roof, the water escapes via that small drain that you referred to. However, this can't handle much water. However, in a storm the water, not including the runoff from the roof, chiefly drains into the lawn that surrounds the backyard.

Third paragraph:

Yes, I am wondering if I should add a new downspout at the point in the last picture where I have circled. I suspect that an additional downspout would help alleviate the burden of water that the one existing downspout has to handle.

The picture below shows the existing downspout at ground level. The downspout attaches to a tube and diverts the water underground and away from the foundation.

Hopefully I am giving you a decent idea as to what's happening.

Install another downspout where you propose but, make this new one the "end of the run" (all the water from the preceding guttering now dumps into this new one). This creates an "stand alone" gutter for your unit. The guttering "run" (from the new downspout to the end of your unit) now dumps into the corner downspout. This run of gutter will need to be re-hung to achieve the proper slope.

Given there's a tree beside your house I'm assuming you've already checked that the gutter isn't overflowing because it's full of debris?

1) Check to make sure there are no clogs in the gutter or downspouts.

2) You should see evidence if the gutter is overflowing by a line where the overflow water is hitting the ground.

3) The idea of diverting water away from the building is a good one. However, the "tube" you are using looks to be a much smaller diameter that the downspout itself thereby backing up the flow. Landscape/irrigation companies usually carry downspout diverters that tightly fit around the downspout. I also suggest getting a lawn "pop up drain" to help prevent clogging. Just an example as there are multiple options . . . https://www.amazon.com/Amerimax-InvisaFlow-Downspout-Strainer-Filter/dp/B07QRS7KVF.

4) When there is rain driven wind, that too can cause water to puddle next to the house, especially a two story building. That should be evaluated. Having sloping steps and possibly sloping ground only makes this situation harder to remedy. Even if the gutter starts functioning to its fullest capacity, that may not correct water collecting in those sloped areas.