The key thing to realize here is that we use the term dynamic range in two related but different ways. First, we use it to describe the scene — the difference between the deepest shadows and the brightest highlights in the actual world before us. Second, we use it to refer to a property of a camera sensor (or film, or print). That is the difference between dark and light that that specific camera (or film, or print) can show. Because technology isn't perfect, this is always limited, and often much more limited than the real world.

So, it isn't that the film or sensor (or printed paper) is magically losing detail when the dynamic range is high. It's that the medium can only capture detail in a certain range, but by adjusting exposure parameters (aperture, shutter speed), you can "slide" that range up and down.

If the scene is low dynamic range, its entirety can be captured, no problem.

But, in a high dynamic range scene, to capture shadow detail, you might slide it down so far that the highlights just can't fit — or, to capture highlights, you might slide the "exposure window" up, and then it can't cover the shadow detail.

The issue is that the exposure adjustments are global — you can't easily decide that you'll have a shorter shutter speed for the sunlight, and a longer one for the umbrella, to get both right. (Although, of course, there are techniques to work around that, as explained in a lot of other questions here — for example How to adjust exposure for both the sky and the landscape?.)

In your first example, you are losing detail on the red basket because, presumably, you've adjusted the exposure to cover the brightly-light rest of the scene. If you add a black umbrella and keep the exposure the same, nothing will change in the rest of the image. However, since that umbrella is likely to be even darker than the basket, it will show even less detail. If you adjust your exposure to compensate for that, showing more shadow detail in both the umbrella and the basket, you will necessarily also increase the exposure of the sky, and if that's outside of the dynamic range of your film or sensor, you'll lose detail there. Note that if you're using automatic exposure, this may happen automatically, as the camera attempts to balance the scene (even cameras which aren't "smart" can do this — a basic exposure program tries to render the whole scene as neutral gray on average, and adding a big area of black will make it choose a brighter exposure.)

In your second example: it's not. You just can't cover them both. Image you're sharing a king-sized bed and have only a narrow blanket. If both you and your friend are at the extreme edges of the bed (by analogy, one very dark and one very light), the blanket can keep one of you warm, but you have to pick which one.