Confusion on Redstone Wire Transmission

According to the wiki on Redstone Circuits:

• Redstone Wire powers itself, the block below it, or the block it is pointing to.

I'm having trouble understanding what this is saying. According to the statement, it sounds like only a one of those three things is ever true at once per redstone wire. However, from my testing it appears redstone can power both the block it is in and the block below it at the same time. I've also noticed that the block below redstone wire can activate a piston next to it (or a redstone torch), but not a redstone wire running next to it (i.e., if you have a block between preventing the redstone wire from running down from the upper block to the lower block).

So what are the conditions that determine whether the block below redstone wire becomes powered? And is there a difference between a "powered" block from redstone wire versus powered blocks from other sources? What am I missing in my mental model of how redstone transmission works?

The confusion lies in the fact that there are two different types of power a block can emit. These are sometimes referred to as strong and weak power.

Weak power is given to a block under the conditions mentioned in the question, by having wire on top of the block or pointing into it. This will unpower torches, extend pistons, play note blocks and turn on lamps & repeaters.

Strong power is given to a block by having a lever on it, a torch underneath it or a repeater pointing into it. In addition to the abilities weak power has, it will also power redstone dust in an adjacent block.