No, these are not equivalent. 1) is the correct way to do it. 2) is missing a Jacobian correction: Since you are doing a non-linear transform of one of the parameters (in the parameter block), you need to add the log determinant of the Jacobi matrix. See, this chapter of the user guide:

A transformation samples a parameter, then transforms it, whereas a change of variables transforms a parameter, then samples it. Only the latter requires a Jacobian adjustment.

When you take care of that, the results of 1) and 2) should be identical. Otherwise, they’re probably still close, but not quite the same.

Alternative 3) is almost the same as 2), but the lower bound on theta_transformed is wrong. If \theta \in (0,\infty), then \log\theta \in (-\infty,\infty), so in this case no (lower) bound is needed.

Also for 3), according to the definition of transformation and change of variables, it seems that 3) is actually the transformation (since I define the transformed parameter) so in this case do I also need to incorporate the Jacobian adjustment or just need to simply correct the lower bound statement?

The easiest way to remember this is looking whether you assign the distribution to a parameter defined in the parameter block (no adjustment needed), or you apply the distribution on a (non-linearly) transformed parameter in the parameters block (adjustment needed).