What is a weakly-simplicial vertex?
According to the 59th slide of the following pdf:

https://grow2015.sciencesconf.org/file/174789 (A talk by D.Kratsch at GROW 2015)

we have that 


  A vertex in a graph is weakly simplicial if its neighborhood is an
  independent set and the neighborhoods of its neighbors form a
  chain under inclusion.


See the slide for an illustration. The context where this definition is found in the slides is chordal bipartite graphs, so I guess this might be what you are looking for.