Frequency response of 2nd order RC low-pass filter
Your answer according to "Method 1" would be correct if you defined \$Z_1\$ as

$$Z_1=\frac{1}{s C_1}||(R_2+\frac{1}{s C_2})=\frac{\frac{1}{s C_1}(R_2+\frac{1}{s C_2})}
{\frac{1}{s C_1}+R_2+\frac{1}{s C_2}}=\frac{1+sR_2C_2}{s^2R_2C_1C_2+s(C_1+C_2)}\tag{1}$$ 

Use Eq. (1) combined with the relation between \$V_0\$ and \$V_1\$ and you will obtain the correct transfer function.