I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability.
I pursue only intuition; do not answer with formal proofs.
Which region in my 2D Venn Diagram below matches $Pr(A|B)$?
On the left, the green = $\Pr(A\cap B)$. The right is herefrom, but fails to indicate or specify $Pr(A|B)$.
The following answers your explicit question: "Which region in my 2D Venn Diagram matches $Pr(B|A)$?"
No single region within the diagram corresponds to $Pr(B|A)$. Instead, $Pr(B|A)$ is the ratio of the green region to the full region of $A$. This term is read: "The probability that $B$ is the case given that A is the case. So... read that phrase "backwards": Given $A$ is the case means you're restricted to be within the disk labeled $A$. Now that you're in that disk, what is the probability that you are within B? That is the ratio of the area of the green region to the area of all of $A$.