Does the Vogel algorithm produce a unique braid? Does the choice of edges of a troubled region on which to perform R2 change the braid word or is there only one sequence of the braid generators no matter which way you resolve the bad areas but possibly shifting the braid word. What I mean is say we obtained the braid word as 1234, would changing which pair of edges we performed R2 on give something like 3412 or would we obtain something completely different?

Vogel's algorithm does not give a unique braid word (even up to cyclic permutation). A superficial way to get different braid representatives is the two different choices of a way to perform a given Reidemeister 2 move. A less superficial way to get two different braid representatives is when there is a choice of where to perform the Reidemeister 2 moves. See the following example.

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In the example above, there are two different ways to perform Vogel's algorithm on a given link diagram. The pictures on the right are the Seifert circles together with the signed crossing information. The directed red arcs are meant to represent the ray that goes from the inside of the braid to the outside.

The first example yields the braid word $$\sigma_1\sigma_2^3\sigma_3\sigma_2\sigma_1^{-1}\sigma_2\sigma_3^{-1}\sigma_2^2.$$ The second example yields the braid word $$\sigma_1\sigma_2\sigma_3^{-1}\sigma_2^{3}\sigma_1^{-1}\sigma_2^2\sigma_3\sigma_2.$$