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I am trying to calculate PCA loadings of a dataset. The more I read about it, the more I get confused because "loadings" is used differently at many places.

I am using sklearn.decomposition in python for PCA analysis as well as R (using factomineR and factoextra libraries) as it provides easy visualization techniques. The following is my understanding:

1. pca.components_ give us the eigen vectors. They give us the directions of maximum variation.
2. pca.explained_variance_ give us the eigen values associated with the eigen vectors.
3. eigenvectors * sqrt(eigen values) = loadings which tell us how principal components (pc's) load the variables.

Now, what I am confused by is:

1. Many forums say that eigen vectors are the loadings. Then, when we multiply the eigen vectors by the sqrt(eigen values) we just get the strength of association. Others say eigenvectors * sqrt(eigen values) = loadings.

2. Eigen vectors squared tells us the contribution of variable to pc? I believe this is equivalent to var$contrib in R.

3. loading squared (eigen vector or eigenvector*sqrt(eigenvalue) I don't know which one) shows how well a pc captures a variable (closer to 1 = variable better explained by a pc). Is this equivalent of var$cos2 in R? If not what is cos2 in R?

4. Basically I want to know how to understand how well a principal component captures a variable and what is the contribution of a variable to a pc. I think they both are different.

5. What is pca.singular_values_? It is not clear from the documentation.

These first and second links that I referred which contains R code with explanation and the statsexchange forum that confused me.