The two citations do not generally contradict each other and both look to me correct. The only underwork is in
Perhaps you mean sum of squared loadings for a principal component, after rotation one should better drop word "principal" since rotated components or factors are not "principal" anymore, to be rigorous. Also (important!) the second citation is correct only when "factor analysis" is actually PCA method (like it is in SPSS by default) and so factors are just principal components. But the table you present is not after PCA, and I wonder whether they are from the same text and wasn't there some misprint.
In the extraction summary table you display there was 23 variables analyzed. Eigenvalues of their correlation matrix are shown in the left section "Initial eigenvalues". No factors have been extracted yet. These eigenvalues correspond to the variances of Principal components (i.e. PCA was performed), not of factors. Adjective "initial" means "at the initiation point of the analysis" and does not imply that there must be some "final" eigenvalues.
The (default in SPSS) Kaiser rule "eigenvalues>1" was used to decide how many factors to extract, so, 4 factors will come. The "eigenvalues>1" rule is based on PCA's eigenvalues (i.e. the eigenvalues of the intact, input correlation matrix).
Extraction of them was done by Principal axis method and the matrix of loadings obtained. Sums of squared loadings in the matrix columns are the factors' variances after extraction. These values appear in the middle section of your table.
These numbers, generally, should not be called eigenvalues because factor extractions not necessarily are based right on the eigendecomposition of the input data - they are specific algorithms on their own. Even Principal axis method which does involve eigenvalues deal with eigenvalues of a repeatedly "trained" matrix, not an original correlation matrix.
But if you had been doing PCA instead of FA then the 4 numbers in the middle column would have been the 4 first eigenvalues identical to the 4 largest ones on the left: in PCA, no fitting take place and the extracted "latent variables" are the PCs themselves, which eigenvalues are their variances.
In the right section, sums of squared loading after rotaion of the factors are shown. The variances of these new, rotated factors. Please read more about rotated factors (or components), especially footnote 4, and that they are neither "principal" anymore nor this-one-to-that-one correspondent to the extracted ones. After rotation, "2nd" factor, for example, is not "2nd extracted factor, rotated". And it also could have greater variance than the "1st" one.
- No, you can't speak of eigenvalues after rotation. No matter be it
orthogonal or oblique.
- You can't even say - at least should better
avoid - of eigenvalues even after extraction of factors unless
these factors are principal components$^1$. (An instructive example showing confusion similar to yours with ML factor extraction.) Variances of factors are SS loadings, not eigenvalues, generally.
- Rotated factors don't correspond one-to-one to the extracted ones.
- The % of total variation
explained by the factors is 40.477% in your example, not 50.317%.
The first number is less because FA factors explain all the assumed
common variation which is less than the portion of total
variation skimmed by the same number of PCs. May say in your report, "The 4-factor solution is responsible for the common variance constituting 40.5% of the total variance; while 4 principal components would account for 50.3% of the total variance".
$^1$ (Before factor rotation) variances of factors (pr. components) are the eigenvalues of the correlation/covariance matrix of the data if the FA is PCA method; variances of factors are the eigenvalues of the reduced correlation/covariance matrix with final communalities on the diagonal, if the FA is PAF method of extraction; variances of factors do not correspond to eigenvalues of correlation/covariance matrix in other FA methods such as ML, ULS, GLS (see). In all cases, variances of orthogonal factors are the SS of the extracted/rotated - final - loadings.