Legendary Banner Pity Rate

Yes and no.

First, to get it out of the way, a point about terminology. It's a bit weird to say the pity rate is 10% when the 5* rate on a legendary banner is always 8% or higher. Nobody's rate drops 10% on a 5* pull unless they managed to go 100 summons without a 5*.

But to the point. Let's say your 5* rate is 10%. If you put a blindfold on and pick any stone, the probability is indeed 1 in 10. However, the 3* and 4* pools are skewed in favour of reds and against greens so that those colours appear a lot more often with non-5*. This is why green stones are relatively rare and yield more 5*s in fewer stones. Fewer 3* and 4* units are showing green, so when you do pull a green, the probability is higher that it's a 5*. In contrast, lots of 3* and 4* stones appear red so in the long run you will pull more 3* and 4* reds for every 5* red. And the same is true on all banners at any appearance rate, but it's especially bad on 3% banners with four focus units of which only one is red because more than a quarter of the off-focus pool is red so it's actually easier to get an off-focus unit than a focus unit.

Yes, the rate is shared by all the stones on the same summoning session, so in case your focus rate is at 10%, then each stone has 1/10 chance of being a focus unit.

I don't know what Seeker meant to say with his first paragraph, after 100 summons with no 5*s the 5* rate changes to 100% guaranting the next season will have 5 5* in it. 10% is pretty common in legendary banners.

Regarding his second paragraph, his reasoning only makes sense if you are sniping a specific color in normal banners where the off-focus 5* exist. In legendary banners, every color has the same chance to give you a 5* regardless of how crowded the 3-4* pool is.