Why do we need difference of radius when calculating the intersection of 2 circles?

Question

While I was looking for how to detect if 2 circles $(x_0, y_0, R_0)$ and $(x_1, y_1, R_1)$ collide I found this answer where it is given that 2 circles collide if and only if

$(R_0 - R_1)^2 \Leftarrow (x_0 - x_1)^2 + (y_0 - y_1)^2 \Leftarrow (R_0 + R_1)^2$.

Why do we need the difference of radius? From my drawing I understand that having only the sum is sufficient:

NOTE: I know the distance in the 3rd representation is not correctly drawn

NOTE2: I couldn't understand cHao's answer

EDIT: Thank you both Netravat Pendsey and Bram28 for your answers. I found out where I was thinking wrong: what I was needing was to see if 2 discs collide, but I mismatched (again) the definition of circle and disc. For disc i think it will be enough only the sum

Answer 1

These circles are colliding:

But these are not: