Neglecting certain elements in deriving the current of a series RLC circuit
You are basically right. But the phase shift would not change anything in the equations (except the last one). The derivative of sin() is still cos() and the relative phase relation stays the same, no matter what phase sin() has. You can do the math yourself. 
cos() is removed in the last equation because cos(t=0) is 1. But if you bring your phase shift in play, then it needs to be considered with cos(phi).