The key result connecting the two is the first equation on page 2, second column. That is literally stick breaking for the intermediate variables z_i = \sin^{2} \theta_i; that said technically the z_i are the complementary proportions used in the stick breaking, so 1 - z_i is the proportion allocated to the $i$th element of the simplex and z_i is the proportion that remains. One can equivalently take v_i = 1 - z_i.

The z_i are constrained to the unit interval but taking a logit completely unconstrains them,

w_i = \text{logit} z_i = \text{logit} ( \sin^{2} (\theta_i) ) \in (-\infty, \infty).

Working the other way one can start with the I - 1 unconstrained, real-valued variables w_i and then construct the z_i,

z_i = \text{logistic} \, w_i

and then use those as incremental complementary proportions,

x_i = (1 - z_i) \prod_{i' = 1}^{i - 1} z_i'.

Equivalently one can add use incremental proportions directly with

v_i = 1 - \text{logistic} \, w_i

and then use those as incremental complementary proportions,

x_i = v_i \prod_{i' = 1}^{i - 1} (1 - v_i').

Note that the “Cruising the Simplex” paper works with the hyperspherical parameters directly and doesn’t actually unconstrain them. The freedom in how to unconstrain them can be used to generate a transformation exactly equivalent to what is used in Stan or something complementary to what is used in Stan. This is also assuming that the ordering of the variables is fixed which is another degree of freedom in both methods.

I’ve mentioned this a few times in other threads but let me mention again that the freedom in constraining/unconstraining transformations is a bit overblown. Any two bijective transformations between a constrained space X and an unconstrained space Y are related by another bijection on Y (i.e. a reparameterization of Y). More formally if \phi_1 : X \rightarrow \mathbb{R} and \phi_2 : X \rightarrow \mathbb{R} then \phi_1 = \psi \circ \phi_2 where \psi : \mathbb{R} \rightarrow \mathbb{R}.

The question of which transformation is “best” is equivalent to asking which parameterization of the unconstrained space is “best” which is depends on the target distribution and is practically intractable.