I liked the mentioned article by Massey very much too, but I guess the best descriptions are still Deligne's SGA 7, part II, chapters 13, 14, where he even draws some figures (BTW, my impression from those chapters and some formulations in other articles by Deligne is that he thought about generalisations, apparently never published). I found Wall: ["Periods of Integrals and Topology of Algebraic Varieties"][1], Griffiths ["summary"][2]
and his ["Periods III"][3] very helpful, parts of this [motivic book][4] too. [Arthur Ogus][5] studies vanishing/nearby cycles in the context of log-geometry. Related to the theme is the local monodromy theorem, about that, monodromy-weight conjecture etc., I found Illusie's article in Asterisque 223 very good.   


  [1]: https://doi.org/10.1098/rspa.1984.0012 "JSTOR:2397500"
  [2]: https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-76/issue-2/Periods-of-integrals-on-algebraic-manifolds--Summary-of-main/bams/1183531477.full "Griffiths summary"
  [3]: http://www.numdam.org/item/?id=PMIHES_1970__38__125_0 "Griffiths Periods III"
  [4]: https://www-fourier.ujf-grenoble.fr/~peters/Books/motivic.f/Tatalects.pdf "Peters Tata book"
  [5]: https://math.berkeley.edu/~ogus/ "Ogus' homepage"