Confusion over current divider rule
For two resistors R1 and R2 in parallel the current divider rule is
$$
I_1 = I \cdot \frac{R_2}{R1+R_2}
$$
where I is the total current and I1 the current through R1.

For two resistors the current through one resistor is the total current times the current of the other resistor divided by the sum of the resistors.

The proof is trivial but here it is:

For two parallel resistors with the same voltage V across them the current through one of them is given by
$$
I_1 = \frac{V}{R_1} = I \cdot \frac{R_1 R_2}{R_1 + R_2} \cdot \frac{1}{R_1}= I \cdot \frac{R_2}{R_1 + R_2} 
$$
since V can be expressed as the product of the total current I times the resistance of R1 and R2 in parallel.

In your first example the total current is I3. The current I1 flows through R1, the other resistor is R2+Ri, the sum is R1+R2+Ri  which results in 
$$I_1=I_3\frac{R_2+R_i}{R_1+R_2+R_i}$$