Spoke washers are used at the hub end, between the flared out cap and the hub itself. The intention is to take up extra length between the bend and the end, to help prevent the spoke from breaking. Common use is on hubs with thin flanges (for lightweight or because they're cheap) and on heavier loaded bikes to spread the load.
Notice how the lower spoke is snugged closer to the hub? The upper spoke is more likely to break sooner.
The nipple washers are for spreading the load at the rim end of the spoke. You're right that the washers have to fit inside the rim track/valley but that's so they lie flat against the inside of the rim, rather than touching in two places and focusing the stresses. Again, touring bikes or load bikes would be the place to find these. Race bikes wouldn't have them.
Nipple washers are used to allow the spoke to leave the rim at a strange angle. Normally the spoke leaves slightly away from 90 degrees (pardon the pun) so the nipple is not flat against the rim, and the spoke is under a slight bending load.
Here's a rim with large domed washers that let the spoke stay straight. I think this might be a motorcycle rim, but the concepts are the same.
If you have a damaged spoke hole, a washer may help, but its a judgement call about how much damage you can safely ride on. If the spoke nipple fits through the nipple hole, I'd not ride on it even with a washer, because all you've done is put the load on a stamped washer then onto the edge of a damaged hole in the rim.
Would you trust this rim if it had a 2c flat washer below the cracked piece? A curved rim washer would just make it worse.
Tubeless tyres take away the worry of pressing on the washer - here's an extreme example on a fatbike wheel: