Stan does neither of these two things. I’ll explain in a moment, but first an aside:
Your options 1 and 2 sample from the same distribution. However, neither option is equivalent to a half normal, because the truncation happens at zero but the normal distribution is centered at 1. Perhaps an error related to this misunderstanding
explains why your simulations seem to yield different distributions for 1 and 2.
So what does Stan do?
Stan imposes the constraint by constructing a new variable under the hood which for illustrative purposes we can call alpha_unconstrained, and then recovering alpha as exp(alpha_unconstrained). Note that alpha is now guaranteed to be positive. Stan then increments the target density according to your sampling statement alpha ~ normal(1, 5); and automatically adds the Jacobian adjustment associated with the constraining transform exp. This results in sampling from the truncated normal distribution obtained by both your procedure 1 and your procedure 2.