Confused about XORing and addition modulo $2$
The confusion here stems from a missing word. A correct statement is "The result of XORing two bits is the same as that of adding those two bits mod 2." 

For example, $(0+1)\bmod 2 = 1\bmod 2 = 1=(0\text{ XOR }1)$ 

and 

$(1+1) \bmod 2= 2\bmod 2 = 0 =(1\text{ XOR }1)$