What does the one function $\mathbf{1}_{i,y^{(t)}}$ exactly mean in backward propagation of RNN in the book "Deep learning" of Bengio
The authors or the publisher may have corrected the inconsistent notation. The current version online looks like below:


Here the $\mathbf{1}_{\text{condition}}$ is the indicator function, as you mentioned.

Regarding the two version of softmax, use the first one for calculating gradient. The second version is for improving numerical stability, and mathmatically it should yield the exact same value as the first version.