phase shift in LRC circuit
Your question is in error. You have fixated on the Theta term, but that is not your problem.

You present two circuits. The first is a simple RLC circuit. You present a circuit analysis of that simple circuit which neglects initial conditions (in particular the initital current through the inductor).


THEN, you show a second circuit which is not equivalent to the first, and you are concerned because the simulation result does not match the calculations and your own simulation based on the FIRST circuit which is different.


Because of the circuit differences, you should have no expectation that the simulation you calculated yourself and the actual simulation results should match. Why don't you try simulating a simple RLC circuit and see if your calculated results match the simulation result when you assume the phase at t=0 is 0?