How to correctly use mask_zero=True for Keras Embedding with pre-trained weights?

You're second approach is correct. You will want to construct your embedding layer in the following way

embedding = Embedding(
   output_dim=embedding_size,
   input_dim=vocabulary_size + 1,
   input_length=input_length,
   mask_zero=True,
   weights=[np.vstack((np.zeros((1, embedding_size)),
                       embedding_matrix))],
   name='embedding'
)(input_layer)

where embedding_matrix is the second matrix you provided.

You can see this by looking at the implementation of keras' embedding layer. Notably, how mask_zero is only used to literally mask the inputs

def compute_mask(self, inputs, mask=None):
    if not self.mask_zero:
        return None
    output_mask = K.not_equal(inputs, 0)
    return output_mask

thus the entire kernel is still multiplied by the input, meaning all indexes are shifted up by one.