init refers to the initial parameter values that Stan starts HMC/NUTS off with. If these values don’t produce a finite log posterior, then Stan has to choose new initial values until it gets a good starting point with a finite log posterior. If no initializer function or list is specified for rstan::stan(), then Stan will sample uniformly on the unconstrained support from the interval (-2, 2) using pseudo RNG. Pseudo RNG is deterministic in the sense that if you provide the same seed, you will get the same sample. So changing the seed will yeild a different set of initial values that may or may not have a finite log posterior. See pg 42 of the rstan manual for more details.
Another approach would be specifying a list or function that returns initial parameter values that you know will yield a finite log posterior. If you do this you should use different initial values for each Markov Chain to get more robust diagnostics.