I can't understand from your first example what, exactly, is going on.
But if you reduce the analysis to AN INTEREST in something ... (whether it's half or full or 1/3) ... then you look at how THAT INTEREST was received to know what the basis is IN that INTEREST.
If I buy a piece of land with by brother (and let's say it's jointly) and my brother dies (and in joint tenancy I am an heir to his half), then I now have a two different bases in the property;
In 1/2 of it, I have my purchase price and in the other half I have a basis of whatever the value was on his date of death.
So let's say we bought for 100,000 (that gives us each as basis of 50,000) and when he dies its worth 200000, ... I now have a basis of 50,000 PLUS the 100,000 basis that was left to me; $150,000 in the property.
NOW, if he had gifted it to my (by quit-claim or any other re-titling, AND I gave him nothing for it, making that a GIFT, then my basis would be 100,000 (my original basis plus his original 50,000).
So, if I sold the property for $300,000 the result would be the following:
... a $150,000 gain, if he had died and left his 1/2/ to me.
... a $200,000 gain, if he gifted it to me.
Think about it this way ... each person does their own taxes, so what matters is not the basis in the asset, but rather that PERSON'S basis in the asset .. based on how much of it they own.
Hope this helps