Those flatter, thinner clouds are less opaque.

In general, shadows from the cloud itself or other clouds explain most of the variation in brightness. But in this case I believe you give the answer yourself when you mention that the flatter clouds should be more translucent:

• they let more of the sunlight through, i.e., they reflect less light and are thus less bright, especially in contrast to the thicker clouds;
• you see them from a shallow angle, with the sun behind you, so most of the light that does get reflected, might be so away from your eyes$^1$;
• also, you're seeing them from above, and they might let you see more of the darker surface underneath.

$^1$Notice how the brightest parts of the clouds seem to be those facing you.

Those flatter, thinner clouds are less opaque.

In general, shadows from the cloud itself or other clouds explain most of the variation in brightness. But in this case I believe you give the answer yourself when you mention that the flatter clouds should be more translucent:

• they let more of the sunlight through, i.e., they reflect less light and are thus less bright, especially in contrast to the thicker clouds;
• you see them from a shallow angle, with the sun behind you, so most of the light that does get reflected, might be so away from your eyes$^1$;
• also, you're seeing them from above, and they might let you see more of the darker surface underneath.

$^1$Notice how the brightest parts of the clouds seem to be those facing you. Edit: As can be seen in the picture below, especially in the highlighted selection, the farther away you look from the line of sight to the sun, the less of its light is reflected toward your eyes.

source: https://cauldronsandcupcakes.com/2012/02/15/wanderlust-and-the-universal-ordering-system/