No, Nothing to do with the temperature. It changes the hardness of the light.

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There are two usages of a white umbrella. As a bounced light inside it, or as a fast-setup softbox, passing the light through it. It is definitely a diffuse light.

The difference versus a softbox is that on a softbox you do not spill light all over the place. When using a white umbrella as a bounce light you can add a black cover so the light does not spill behind it, you have still much light all over the place. It is a good option if you want for example high key portraits.

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But a silver umbrella is only used as a bounce light. Depending on the overall size, and the position of the flash inside it, it can produce different results.

One example is parabolic umbrellas. If the umbrella is a real parabolic one, (big umbrellas and specialized brands) can give you more or less parallel rays, where the fall-off of the light is lower than a normal studio light. The light should be on the focal point.

But a white one will spill more light, regardless of the shape of the umbrella.

In general, is due to the nature of how light is bounced on different surfaces.

The diagrams are exaggerated, especially the silver ones.

That would be a perfect mirror, like a chromed metallic surface. The finish of silver umbrellas has some diffusion but not as much as white.