Two ways to find the "distance" between encodings

No, those two methods give different results. The second method will give you the square root of the first method. Do you know the definition of the L2 norm of a vector? It is the traditional Euclidean length of the vector:

||v|| = \displaystyle \sqrt { \sum_{i = 1}^{n} v_i^2}

As to why they use the square in the triplet_loss case and the plain L2 norm in the verify case, I don’t know. I think it’s just a choice they have made.