Latent Semantic Indexing coordinates vs PCA scores

For PCA:

Let $$X$$ be a mean-centered matrix having as columns observations and as rows features. Via SVD, we have $$X = U \Sigma V^\top$$. Then PCA scores are $$U^\top X = \Sigma V^\top$$.

For LSA/LSI:

Let $$X$$ be a term-document matrix having as columns documents and as rows terms. Via SVD, we have $$X = U \Sigma V^\top$$. Then the new coordinates of the documents in the latent space are $$\Sigma ^ {-1} U^\top X = V^\top$$, according to wikipedia or this, but they are $$U^\top X = \Sigma V^\top$$, according to this.

My questions:

1. Is there a standard way of computing LSI coordinates? My intuition was that they should be the same as in the PCA case, but Wikipedia changed my mind.

2. If there is no standard, are there any advantages of choosing one over another?